Rolling Factors Deformations

and Extensions of Canonical Curves

Jan Stevens

Received: October 10, 2000

Communicated by Thomas Peternell

Abstract. A tetragonal canonical curve is the complete intersection of two divisors on a scroll. The equations can be written in ‘rolling factors’ format. For such homogeneous ideals we give methods to compute infinitesimal deformations. Deformations can be obstructed. For the case of quadratic equations on the scroll we derive explicit base equations. They are used to study extensions of tetragonal curves.

2000 Mathematics Subject Classification: 14B07 14H51 14J28 32S30 Keywords and Phrases: Tetragonal curves, rolling factors, K3 surfaces

An easy dimension count shows that not all canonical curves are hyperplane sections of K3 surfaces. A surface with a given curve as hyperplane section is called an extension of the curve. With this terminology, the general canonical curve has only trivial extensions, obtained by taking a cone over the curve. In this paper we concentrate on extensions of tetragonal curves. The extension problem is related to deformation theory for cones. This is best seen in terms of equations. Suppose we have coordinates (x_{0} : ⋯: x_{n} : t) on ℙ^{n+1} with the special hyperplane section given by t = 0. We describe an extension W of a variety V: f_{j}(x_{i}) = 0 by a system of equations F_{j}(x_{i}, t) = 0 with F_{j}(x_{i}, 0) = f_{j}(x_{i}). We write F_{j}(x_{i}, t) = f_{j}(x_{i}) + tf′_{j}(x_{i}) + ⋯ + a_{j}t^{d_{j}}, where d_{j} is the degree of F_{j}. Considering (x_{0}, …, x_{n}, t) as affine coordinates on ℂ^{n+1} × ℂ we can read the equations in a different way. The equations f_{j}(x_{i}) = 0 define the affine cone C(V) over V and F_{j}(x_{i}, t) = 0 describes a 1parameter deformation of C(V). The corresponding infinitesimal deformation is f_{j}(x_{i}) ↦ f′_{j}(x_{i}), which is a deformation of weight −1. Conversely, given a 1-parameter deformation F_{j}(x_{i}, t) = 0 of C(V), with F_{j} homogeneous of degree d_{j}, we get an extension W of V. For most of the cones considered here the only infinitesimal deformations of negative weight have weight −1 and in that case the versal deformation in negative weight gives a good description of all possible extensions.

As the number of equations typically is much larger than the codimension one needs good ways to describe them. A prime example is a determinantal scheme X: its ideal is generated by the t × t minors of an r × s matrix, which gives a compact description of the equations. Following Miles Reid we call this a format. Canonical curves are not themselves determinantal, but they do lie on scrolls: a k-gonal curve lies on a (k − 1)-dimensional scroll, which is given by the minors of a 2 × (g − k + 1) matrix. For k = 3 the curve is a divisor on the scroll, given by one bihomogeneous equation, and for k = 4 it is a complete intersection, given by two bihomogeneous equations. In these cases there is a simple procedure (‘rolling factors’) to write out one resp. two sets of equations on ℙ^{g−1} cutting out the curve on the scroll. Powerful methods exist to compute infinitesimal deformations without using explicit equations. We used them for the extension problem for hyperelliptic curves of high degree [Stevens 1996] and trigonal canonical curves [Drewes– Stevens 1996]. In these papers also several direct computations with the equations occur. They seem unavoidable for tetragonal curves, the subject of a preprint by James N. Brawner [Brawner 1996]. The results of these computations do not depend on the particular way of choosing the equations cutting out the curve on the scroll. This observation was the starting point of this paper. We distinguish between different types of deformations and extensions. If only the equations on the scroll are deformed, but not the scroll itself we speak of pure rolling factors deformations. A typical extension lies then on the projective cone over the scroll. Such a cone is a special case of a scroll of one dimension higher. If the extension lies on a scroll which is not a cone, the equations of the scroll are also deformed. We have a rolling factors deformation. Finally if the extension does not lie on a scroll of one dimension higher we are in the situation of a non-scrollar deformation. Non-scrollar extensions of tetragonal curves occur only in connection with Del Pezzo surfaces. Not every infinitesimal deformation of a scroll gives rise to a deformation of complete intersections on it. One needs certain lifting conditions, which are linear equations in the deformation variables of the scroll. Our first main result describes them, depending only on the coefficients of the equations on the scroll. The next problem is to extend the infinitesimal deformations to a versal deformation. Here we restrict ourselves to the case that all defining equations are quadratic. Our methods thus do not apply to trigonal curves, but we can handle tetragonal curves. Rolling factors obstructions arise. Previously we observed that one can write them down, given explicit equations on ℙ^{n} [Stevens 1996, Prop. 2.12]. Here we give formulas depending only on the coefficients of the equations on the scroll. As first application we study base spaces for hyperelliptic cones. The equations have enough structure so that explicit solutions can be given. Surfaces with canonical hyperplane sections are a classical subject. References to the older literature can be found in Epema’s thesis [Epema 1983], which is especially relevant for our purposes. His results say that apart from K3 surfaces only rational surfaces or birationally ruled surfaces can occur. Furthermore he describes a construction of such surfaces. Extensions of pure rolling factors type of tetragonal curves fit very well in this description. A general rolling factors extension is a complete intersection on a nonsingular four-dimensional scroll. The classification of such surfaces [Brawner 1997], which we recall below, shows that surfaces with isolated singularities and in particular K3s can only occur if the degrees of the equations on the scroll differ at most by 4. A tetragonal curve of high genus with general discrete invariants has no pure rolling factors deformations. Extensions exist if the base equations have a solution. For low genus we have more variables than equations. For the maximal genus where almost all curves have a K3 extension we find:

Proposition. The general tetragonal curve of genus 15 is hyperplane section of 256 different K3 surfaces.

We also look at examples with genus 16 and 17. It is unclear to us which property of a curve makes it have an extension (apart from the property of being a hyperplane section). The contents of this paper is as follows. First we describe the rolling factors format and explain in detail the equations and relations for the complete intersection of two divisors on a scroll. Next we recall how canonical curves fit into this pattern. In particular we describe the discrete invariants for tetragonal curves. The same is done for K3 surfaces. The second section is devoted to the computation of infinitesimal deformations. First non-scrollar deformations are treated, followed by rolling factors deformations. The main result here describes the lifting matrix. As application the dimension of T^{1} is determined for tetragonal cones. In the third section the base equations for complete intersections of quadrics on scrolls are derived. As examples base spaces for hyperelliptic cones are studied. The final section describes extensions of tetragonal curves.

1. Rolling factors format.

A subvariety of a determinantal variety can be described by the determinantal equations and additional equations obtained by ‘rolling factors’ [Reid 1989]. A typical example is the case of divisors on scrolls. We start with a k-dimensional rational normal scroll S ⊂ ℙ^{n} (for the theory of scrolls we refer to [Reid 1997]). The classical construction is to take k complementary linear subspaces L_{i} spanning ℙ^{n}, each containing a parametrized rational normal curve ϕ_{i}: ℙ^{1} → C_{i} ⊂ L_{i} of degree d_{i} = dim L_{i}, and to take for each p ∈ ℙ^{1} the span of the points ϕ_{i}(p). The degree of S is d = ∑ d_{i} = n−k+1. If all d_{i} > 0 the scroll S is a ℙ^{d−1}-bundle over ℙ^{1}. We allow however that d_{i} = 0 for some i. Then S is the image of ℙ^{d−1}-bundle S̃ over ℙ^{1} and S̃ → S is a rational resolution of singularities. To give a coordinate description, we take homogeneous coordinates (s : t) on ℙ^{1}, and (z^{(1)} : ⋯: z^{(k)}) on the fibres. Coordinates on ℙ^{n} are zj^{(i)} = z^{(i)}s^{d_{i}−j}t^{j}, with 0 ≤ j ≤ d_{i}, 1 ≤ i ≤ k. We give the variable z^{(i)} the weight −d_{i}. The scroll S is given by the minors of the matrix

Φ = ( \begin{matrix}z0^{(1)} & … & zd_{1}−1^{(1)} & … & z0^{(k)} & … & zd_{k}−1^{(k)} \\ z1^{(1)} & … & zd_{1}^{(1)} & … & z1^{(k)} & … & zd_{k}^{(k)} \end{matrix} ) .

We now consider a divisor on S̃ in the linear system |aH − bR|, where the hyperplane class H and the ruling R generate the Picard group of S̃. When we speak of degree on S̃ this will be with respect to H. The divisor can be given by one bihomogeneous equation P (s, t, z^{(i)}) of degree a in the z^{(i)}, and total degree −b. By multiplying P (s, t, z^{(i)}) with a polynomial of degree b in (s:t) we obtain an equation of degree 0, which can be expressed as polynomial of degree a in the zj^{(i)}; this expression is not unique, but the difference of two expressions lies in the ideal of the scroll. By the obvious choice, multiplying with s^{b−m}t^{m}, we obtain b + 1 equations P_{m}. In the transition from the equation P_{m} to P_{m+1} we have to increase by one the sum of the lower indices of the factors zj^{(i)} in each monomial, and we can and will always achieve this by increasing exactly one index. This amounts to replacing a zj^{(i)}, which occurs in the top row of the matrix, by the element zj+1^{(i)} in the bottom row of the same column. This is the procedure of ‘rolling factors’.

Example 1.1. Consider the cone over 2d − b points in ℙ^{d}, lying on a rational normal curve of degree d, with b < d. Let the polynomial P (s, t) = p_{0}s^{2d−b} + p_{1}s^{2d−b−1}t + ⋯ + p_{2d−b}t^{2d−b} determine the points on the rational curve. We get the determinantal

| \begin{matrix}z_{0} & z_{1} & … & z_{d−1} \\ z_{1} & z_{2} & … & z_{d} \end{matrix} |

and additional equations P_{m}. To be specific we assume that b = 2c:

P_{0} = p_{0}z_{0}^{2} + p_{1}z_{0}z_{1} + ⋯ + p_{2d−2c−1}z_{d−c−1}z_{d−c} + p_{2d−2c}z_{d−c}^{2}
P_{1} = p_{0}z_{0}z_{1} + p_{1}z_{1}^{2} + ⋯ + p_{2d−2c−1}z_{d−c}^{2} + p_{2d−2c}z_{d−c}z_{d−c+1}
⋮
P_{2c} = p_{0}z_{c}^{2} + p_{1}z_{c}z_{c+1} + ⋯ + p_{2d−2c−1}z_{d−1}z_{d} + p_{2d−2c}z_{d}^{2} .

The ‘rolling factors’ phenomenon can also occur if the entries of the matrix are more general.

Example 1.2. Consider a non-singular hyperelliptic curve of genus 5, with a half-canonical line bundle L = g_{1}^{2} + P_{1} + P_{2} where the P_{i} are Weierstrass points. According to [Reid 1989], Thm. 3, the ring R(C, L) = ⨁ H^{0}(C, nL) is k[x_{1}, x_{2}, y_{1}, y_{2}, z_{1}, z_{2}]/I with I given by the determinantal

| \begin{matrix}x_{1} & y_{1} & x_{2}^{2} & z_{1} \\ x_{2} & x_{1}^{2} & y_{2} & z_{2} \end{matrix} |

and the three rolling factors equations

\begin{aligned}z_{1}^{2} & = x_{1}^{2}h + y_{1}^{3} + x_{2}^{4}y_{2} \\ z_{1}z_{2} & = x_{1}x_{2}h + y_{1}^{2}x_{1}^{2} + x_{2}^{2}y_{2}^{2} \\ z_{2}^{2} & = x_{2}^{2}h + y_{1}x_{1}^{4} + y_{2}^{3} \end{aligned}

where h is some quartic in x_{1}, x_{2}, y_{1}, y_{2}.

The description of the syzygies of a subvariety V of the scroll S proceeds in two steps. First one constructs a resolution of O_{V} by vector bundles on S̃ which are repeated extensions of line bundles. Schreyer describes, following Eisenbud, Eagon-Northcott type complexes C^{b} such that C^{b}(a) is the minimal resolution of i_{∗}(O_{S}(−aH + bR)) as O_{ℙ^{n}}-module, if b ≥ −1 [Schreyer 1986]. Here i: S̃ → ℙ^{n} is the map defined by H. The resolution of O_{V} is then obtained by taking an (iterated) mapping cone. The matrix Φ defining the scroll can be obtained intrinsically from the multiplication map

H^{0}O_{S}(R) ⊗ H^{0}O_{S}(H − R) ⟶ H^{0}O_{S}(H) .

In general, given a map Φ: F → G of locally free sheaves of rank f and g respectively, f ≥ g, on a variety one defines Eagon-Northcott type complexes C^{b}, b ≥ −1, in the following way:

C_{j}^{b} = \{ \begin{matrix}⋀^{j} F ⊗ S_{b−j}G, & for 0 ≤ j ≤ b \\ ⋀^{j+g−1} F ⊗ D_{j−b−1}G^{∗} ⊗ ⋀^{g} G^{∗}, & for j ≥ b + 1 \end{matrix}

with differential defined by multiplication with Φ ∈ F^{∗} ⊗ G for j ≠ b + 1 and ⋀^{g} Φ ∈ ⋀^{g} F^{∗} ⊗ ⋀^{g} G for j = b + 1 in the appropriate term of the exterior, symmetric or divided power algebra. In our situation F ≅ O_{ℙ^{n}}^{d}(−1) and G ≅ O_{ℙ^{n}}^{2} with Φ given by the matrix of the scroll. Then C^{b}(−a) is for b ≥ −1 the minimal resolution of O_{S}(−aH + bR) as O_{ℙ}-module [Schreyer 1986, Cor. 1.2]. Now let V ⊂ S ⊂ ℙ^{n} be a ‘complete intersection’ of divisors Y_{i} ∼ a_{i}H − b_{i}R, i = 1, …, l, on a k-dimensional rational scroll of degree d with b_{i} ≥ 0. The resolution of O_{V} as O_{S}-module is a Koszul complex and the iterated mapping cone of complexes C^{b} is the minimal resolution [Schreyer 1986, Sect. 3, Example]. To make this resolution more explicit we look at the case l = 2, which is relevant for tetragonal curves. The iterated mapping cone is

[C^{b_{1}+b_{2}}(−a_{1} − a_{2}) ⟶ C^{b_{1}}(−a_{1}) ⊕ C^{b_{2}}(−a_{2})] ⟶ C^{0}

To describe equations and relations we give the first steps of this complex. We first consider the case that b_{1} ≥ b_{2} > 0. We write O for O_{ℙ^{n}}. We get the double complex

O  ⟵ ⋀^{2} O^{d}(−1)  ⟵  ⋀^{3} O^{d}(−1)⊗O^{2}
↑  ↑
S_{b1} O^{2}(−a_{1})⊕S_{b2} O^{2}(−a_{2}) ⟵ O^{d}(−1)⊗S_{b1−1}O^{2}(−a_{1})⊕O^{d}(−1)⊗S_{b2−1}O^{2}(−a_{2})
↑
S_{b1+b2} O^{2}(−a_{1}−a_{2})

The equations for V consist of the determinantal ones plus two sets of additional equations obtained by rolling factors: the two equations P^{(1)}, P^{(2)} defining V on the scroll give rise to b_{1} + 1 equations Pm^{(1)} and b_{2} + 1 equations Pm^{(2)}. To describe the relations we introduce the following notation. A column in the matrix Φ has the form (zj^{(i)}, zj+1^{(i)}). We write symbolically (z_{α}, z_{α+1}), where the index α stands for the pair j^{(i)} and α + 1 means adding 1 to the lower index. More generally, if α = j^{(i)} and α′ = j′^{(i′)} then the sum α + α′ := j + j′ only involves the lower indices. To access the upper index we say that α is of type i. The rolling factors assumption is that two consecutive additional equations are of the form

\begin{matrix}P_{m} = & ∑_{α} p_{α,m}z_{α}, \\ P_{m+1} = & ∑_{α} p_{α,m}z_{α+1}. \end{matrix}

where the polynomials p_{α,m} depend on the z-variables and the sum runs over all possible pairs α = j^{(i)}. To roll from P_{m+1} to P_{m+2} we collect the ‘coefficients’ in the equation P_{m+1} in a different way: we also have P_{m+1} = ∑_{α} p_{α,m+1}z_{α}. We write the scrollar equations as f_{αβ} = z_{α}z_{β+1} − z_{α+1}z_{β}. The relations between them are

R_{α,β,γ} = f_{α,β}z_{α} − f_{α,γ}z_{β} + f_{β,γ}z_{α},
S_{α,β,γ} = f_{α,β}z_{α+1} − f_{α,γ}z_{β+1} + f_{β,γ}z_{α+1},

which corresponds to the term ⋀^{3} O^{d}(−1) ⊗ O^{2} in Schreyer’s resolution. The second line yields relations involving the two sets of Pm^{(n)} :

R_{β,m}^{n} = Pm+1^{(n)}z_{β} − Pm^{(n)}z_{β+1} − ∑_{α} f_{β,α}pα,m^{(n)},

where n = 1, 2 and 0 ≤ m < b_{i}. We note the following relation:

R_{β,m}^{n}z_{γ} − R_{β,m}^{n}z_{β} − ∑ R_{β,γ,α}^{n}pα,m^{(n)} = Pm^{(n)}f_{β,γ} − f_{β,γ}Pm^{(n)}.

The right hand side is a Koszul relation; the second factor in each product is considered as coefficient. There are similar expressions involving z_{γ+1}, z_{β+1} and S_{β,γ,α}. Finally by multiplication with suitable powers of s and t the Koszul relation P^{(1)}P^{(2)} − P^{(2)}P^{(1)} gives rise to b_{1} + b_{2} + 1 relations — this is the term S_{b_{1}+b_{2}}O^{2}(−a_{1} − a_{2}). In case b_{1} > b_{2} = 0 the resolution is

O  ←  ⋀^{2} O^{d}(−1) ← ⋀^{3} O^{d}(−1) ⊗ O^{2}
↑  ↑
S_{b_{1}}O^{2}(−a_{1}) ⊕ O^{2}(−a_{2}) ← O^{d}(−1) ⊗ S_{b_{1}−1}O^{2}(−a_{1}) ⊕ ⋀^{2} O^{d}(−1 − a_{2})
↑
S_{b_{1}}O^{2}(−a_{1} − a_{2})

The new term expresses the Koszul relations between the one equation P^{(2)} and the determinantal equations (which had previously been expressible in terms of rolling factors relations). For the computation of deformations these relations may be ignored. Finally, if b_{2} = −1, the equations change drastically.

(1.3) Canonical curves [Schreyer 1986]. A k-gonal canonical curve lies on a (k − 1)-dimensional scroll of degree d = g − k + 1. We write D for the divisor of the g_{k}^{1}. To describe the type S(e_{1}, ..., e_{k−1}) of the scroll we introduce the numbers

f_{i} = h^{0}(C, K − iD) − h^{0}(C, K − (i + i)D) = k + h^{0}(iD) − h^{0}((i + 1)D)

for i ≥ 0 and set

e_{i} = #{j | f_{j} ≥ i} − 1 .

In particular, e_{1} is the minimal number i such that h^{0}((i + 1)D) − h^{0}(iD) = k and it satisfies therefore e_{1} ≤ \frac{2g−2}{k}. A trigonal curve lies on a scroll of type S(e_{1}, e_{2}) and degree d = e_{1} + e_{2} = g − 2 with

\frac{2g − 2}{3} ≥ e_{1} ≥ e_{2} ≥ \frac{g − 4}{3}

as a divisor of type 3H − (g − 4)R. The minimal resolution of O_{C} is given by the mapping cone

C^{d−2}(−3) ⟶ C^{0} .

Introducing bihomogeneous coordinates (x : y; s : t) and coordinates x_{i} = xs^{e_{1}−i}t^{i}, y_{i} = ys^{e_{2}−i}t^{i} we obtain the scroll

( \begin{matrix}x_{0} & x_{1} & … & x_{e_{1}−1} & y_{0} & y_{1} & … & y_{e_{2}−1} \\ x_{1} & x_{2} & … & x_{e_{1}} & y_{1} & y_{2} & … & y_{e_{2}} \end{matrix} )

and a bihomogeneous equation for C

P = A_{2e_{1}−e_{2}+2}x^{3} + B_{e_{1}+2}x^{2}y + C_{e_{2}+2}xy^{2} + D_{2e_{2}−e_{1}+2}y^{3}

where A_{2e_{1}−e_{2}+2} is a polynomial in (s:t) of degree 2e_{1} −e_{2} +2 and similarly for the other coefficients. By rolling factors P gives rise to g − 3 extra equations. The inequality e_{1} ≤ \frac{2g−2}{3} can also be explained from the condition that the curve C is nonsingular, which implies that the polynomial P is irreducible, and therefore the degree 2e_{2} − e_{1} + 2 = 2g − 2 − 3e_{1} of the polynomial D_{2e_{2}−e_{1}+2} is nonnegative. The other inequality follows from this one because e_{1} = g−e_{2} −2, but also by considering the degree of A_{2e_{1}−e_{2}+2}. A tetragonal curve of genus g ≥ 5 is a complete intersection of divisors Y ∼ 2H − b_{1}R and Z ∼ 2H − b_{2}R on a scroll of type S(e_{1}, e_{2}, e_{3}) of degree d = e_{1} + e_{2} + e_{3} = g − 3, with b_{1} + b_{2} = d − 2, and

\frac{g − 1}{2} ≥ e_{1} ≥ e_{2} ≥ e_{3} ≥ 0

We introduce bihomogeneous coordinates (x: y: z; s: t). Then Y is given by an equation

P = P_{1,1}x^{2} + P_{1,2}xy + ⋯ + P_{3,3}z^{2}

with P_{ij} (if nonzero) a polynomial in (s: t) of degree e_{i} + e_{j} − b_{1} and likewise Z has equation

Q = Q_{1,1}x^{2} + Q_{1,2}xy + ⋯ + Q_{3,3}z^{2}

with deg Q_{ij} = e_{i} + e_{j} − b_{2}. The minimal resolution is of type discussed above, because the condition −1 ≤ b_{2} ≤ b_{1} ≤ d−1 is satisfied: the only possibility to have a divisor of type 2H −bR with b ≥ d is to have e_{1} = e_{2} = d/2, e_{3} = 0 and b = d, but then the equation P is of the form αx^{2} + βxy + γy^{2} with constant coefficients, so reducible. If b_{2} = −1 also cubics are needed to generate the ideal, so the curve admits also a g_{3}^{1} or g_{5}^{2}; this happens only up to g = 6. We exclude these cases and assume that b_{2} ≥ 0.

Lemma 1.4. We have b_{1} ≤ 2e_{2} and b_{2} ≤ 2e_{3}.

Proof . If b_{1} > 2e_{2} the polynomials P_{22}, P_{23} and P_{33} vanish so P is reducible and therefore C. If b_{2} > 2e_{3} then P_{33} and Q_{33} vanish. This means that the section x = y = 0 is a component of Y ∩ Z on the ℙ^{2}-bundle whose image in ℙ^{g−1} is the scroll (if e_{3} > 0 the scroll is nonsingular, but for e_{3} = 0 it is a cone). As the arithmetic genus of Y ∩ Z is g and its image has to be the nonsingular curve C of genus g, the line cannot be a component. □

This Lemma is parts 2 – 4 in [Brawner 1997, Prop. 3.1]. Its last part is incorrect. It states that b_{1} ≤ e_{1} + e_{3} if e_{3} > 0, and builds upon the fact that Y has only isolated singularities. However the discussion in [Schreyer 1986] makes clear that this need not be the case. The surface Y fibres over ℙ^{1}. There are now two cases, first that the general fibre is a non-singular conic. In this case one of the coefficients P_{13}, P_{23} or P_{33} is nonzero, giving indeed b_{1} ≤ e_{1} + e_{3}.

The other possibility is that each fibre is a singular conic. Then Y is a birationally ruled surface over a (hyper)elliptic curve E with a rational curve \overline{E} of double points, the canonical image of E, and C does not intersect \overline{E}. This means that the section \overline{E} of the scroll does not intersect the surface Z, so if one inserts the parametrisation of \overline{E} in the equation of Z one obtains a non-zero constant. Let the section be given by polynomials in (s: t), which if nonzero have degree d_{s} − e_{1}, d_{s} − e_{2}, d_{s} − e_{3}. Inserting them in the polynomial Q gives a polynomial of degree 2d_{s} − b_{2}. So b_{2} is even and 2d_{s} = b_{2} ≤ 2e_{3}. On the other hand d_{s} − e_{3} ≥ 0 so d_{s} = e_{3} and b_{2} = 2e_{3}. The genus of E satisfies p_{a}(E) = b_{2}/2 + 1. If b_{1} > e_{1} + e_{3}, then Y is singular along the section x = y = 0. An hyperelliptic involution can also occur if b_{1} ≤ e_{1} + e_{3}. We have shown:

Lemma 1.5. If Y is singular, in particular if b_{1} > e_{1} + e_{3}, then b_{2} = 2e_{3}.

Finally we analyze the case b_{2} = 0 (cf. [Brawner 1996]).

Lemma 1.6. A nonsingular tetragonal curve is bi-elliptic or lies on a Del Pezzo surface if and only if b_{2} = 0. The first case occurs for e_{3} = 0, and the second for the values (2, 0, 0), (1, 1, 0), (2, 1, 0), (1, 1, 1), (3, 1, 0), (2, 2, 0), (2, 1, 1), (3, 2, 0), (2, 2, 1), (4, 2, 0), (3, 2, 1) or (2, 2, 2) of the triple (e_{1}, e_{2}, e_{3}).

Proof . If the curve is bielliptic or lies on a Del Pezzo, the g_{4}^{1} is not unique, which implies that the scroll is not unique. This is only possible if b_{2} = 0 by [Schreyer 1986], p. 127. Then C is the complete intersection of a quadric and a surface Y of degree g − 1, which is uniquely determined by C. The inequality e_{1} + e_{2} + e_{3} − 2 = b_{1} ≤ 2e_{2} shows that e_{3} ≤ e_{2} − e − 1 + 2 ≤ 2. If the general fibre of Y over ℙ^{1} is non-singular we have b_{1} ≤ e_{1} + e_{3}. This gives e_{2} ≤ 2 and b_{1} ≤ 4. The possible values are now easily determined. If the general fibre of Y is singular then e_{3} = b_{2}/2 = 0 and Y is an elliptic cone. □

(1.7) K3 surfaces. Let X be a K3 surface (with at most rational double point singularities) on a scroll. If the scroll is nonsingular the projection onto ℙ^{1} gives an elliptic fibration on X, whose general fibre is smooth. This is even true if the scroll is singular: the strict transform X̃ on S̃ has only isolated singularities. We start with the case of divisors. A treatment of such scrollar surfaces with an elliptic fibration can be found in [Reid 1997, 2.11]. One finds:

Lemma 1.8. For the general F ∈ |3H −kR| on a scroll S(e_{1}, e_{2}, e_{3}) the general fibre of the elliptic fibration is a nonsingular cubic curve if and only if k ≤ 3e_{2} and k ≤ e_{1} + 2e_{3}.

If one fixes k and e_{1} + e_{2} + e_{3} these conditions limit the possible distribution of the integers (e_{1}, e_{2}, e_{3}). By the adjunction formula one has k = e_{1}+e_{2}+e_{3}−2 for a K3 surface. In this case we obtain 12 solutions, which fall into 3 deformation types of scrolls, according to ∑ e_{i} (mod 3):

(e + 2, e, e − 2) → (e + 1, e, e − 1) → (e, e, e)
(e + 3, e, e − 2) → (e + 2, e, e − 1) → (e + 1, e + 1, e − 1) → (e + 1, e, e)

(e+4, e, e−2) → (e+3, e, e−1) → (e+2, e+1, e−1) → (e+2, e, e) → (e+1, e+1, e)

The general element of the linear system can only have singularities at the base locus. The base locus is the section (0:0:1) if and only if k > 3e_{3} and there is a singularity at the points (s:t) where both A_{e_{1}+2e_{3}−k} and A_{e_{2}+2e_{3}−k} vanish. The assumption that the coefficients are general implies now that deg A_{e_{2}+2e_{3}−k} < 0 and deg A_{e_{1}+2e_{3}−k} > 0. In the 12 cases above this occurs only for (e + 3, e, e − 1) and (e + 2, e, e − 1). In the first case the term y^{2}z is also missing, yielding that there is an A_{2}- singularity at the only zero of A_{e_{2}+2e_{3}−k}, whereas the second case gives an A_{1}. The scroll S_{e+4,e,e−2} deforms into S_{e+3,e,e−1}, but the general K3-surface on it does not deform to a K3 on S_{e+3,e,e−1}, but only those with an A_{2}-singularity. These results hold if all e_{i} > 0; we leave the modifications in case e_{3} = 0 to the reader. The tetragonal case is given as exercise in [Reid 1997] and the complete solution (modulo some minor mistakes) can be found in [Brawner 1997]. We give the results:

Lemma 1.9. For the general complete intersection of divisors of type 2H −b_{1}R and 2H − b_{2}R on a scroll S_{e_{1},e_{2},e_{3},e_{4}} the general fibre of the elliptic fibration is a nonsingular quartic curve if and only if either α: b_{1} ≤ e_{1} + e_{3}, b_{1} ≤ 2e_{2} and b_{2} ≤ 2e_{4}, or β: b_{1} ≤ e_{1} + e_{4}, b_{1} ≤ 2e_{2}, 2e_{4} < b_{2} ≤ 2e_{3} and b_{2} ≤ e_{2} + e_{4}.

Proposition 1.10. The general element is singular at a point of the section (0 : 0 : 0 : 1) if the invariants satisfy in addition one of the following conditions:

1α: b_{2} < 2e_{4}, b_{1} > e_{1} + e_{4}. 1βi: b_{2} ≤ e_{3} + e_{4}, e_{2} + e_{4} < b_{1} < e_{1} + e_{4}. 1βii: b_{2} > e_{3} + e_{4}, and e_{1} + e_{2} + 2e_{4} > b_{1} + b_{2}. There is a singularity with z ≠ 0 if

2α: b_{1} > e_{2} + e_{3}, e_{1} + e_{3} > b_{1} > e_{1} + e_{4}. 2αβ: e_{1} + e_{4} ≥ b_{1} > e_{2} + e_{3} and

i: if b_{2} ≤ 2e_{4} then 2(e_{1} + e_{3} + e_{4}) > 2b_{1} + b_{2}

ii: if 2e_{4} < b_{2} ≤ e_{3} + e_{4} then e_{1} + 2e_{3} + e_{4} > b_{1} + b_{2}
iii: e_{3} + e_{4} < b_{2} < 2e_{3}

For K3 surfaces we need b_{1} + b_{2} = e_{1} + e_{2} + e_{3} + e_{4} − 2. We give a table listing the possibilities under this assumption, cf. [Brawner 1997, Table A.1–A.4]. The table lists the possible values for (b_{1}, b_{2}) and gives for each pair the invariants (e_{1}, e_{2}, e_{3}, e_{4}) of the scrolls on which the curve can lie. These form one deformation type with adjacencies going vertically, except S_{e+2,e,e,e} and

(b_{1}, b_{2})  (e_{1}, e_{2}, e_{3}, e_{4})  M  base  sings
(2e, 2e − 2)  (e + 3, e + 1, e − 1, e − 3)  17  B_{e−1}  −−
(e + 3, e, e − 1, e − 2)  15  B_{e−1}  A_{3}
(e + 2, e + 1, e − 1, e − 2)  16  B_{e−1}  A_{1}
(e + 2, e, e, e − 2)  16  B_{e−2}  −−
(e + 2, e, e − 1, e − 1)  15  B_{e−1}  2A_{1}
(e + 1, e + 1, e − 1, e − 1)  16  B_{e−1}  −−
(e + 1, e, e, e − 1)  17  B_{e−1}  −−
(e, e, e, e)  17  ∅  −−
(2e − 1, 2e − 1)  (e + 1, e + 1, e, e − 2)  17  B_{e−2}  −−
(e + 1, e, e, e − 1)  17  B_{e−1}  −−
(e, e, e, e)  18  ∅  −−
(2e + 1, 2e − 2)  (e + 4, e + 1, e − 1, e − 3)  17  B_{e−1}  −−
(e + 3, e + 1, e − 1, e − 2)  16  B_{e−1}  A_{1}
(e + 2, e + 1, e − 1, e − 1)  16  B_{e−1}  −−
(e + 1, e + 1, e, e − 1)  17  B_{e}  −−
(2e, 2e − 1)  (e + 2, e + 1, e, e − 2)  17  B_{e−2}  −−
(e + 2, e, e, e − 1)  15  B_{e−1}  A_{1}
(e + 1, e + 1, e, e − 1)  17  B_{e−1}  −−
(e + 1, e, e, e)  18  ∅  −−
(2e + 2, 2e − 2)  (e + 5, e + 1, e − 1, e − 3)  18  B_{e−1}  −−
(e + 4, e + 1, e − 1, e − 2)  17  B_{e−1}  A_{1}
(e + 3, e + 1, e − 1, e − 1)  17  B_{e−1}  −−
(e + 2, e + 1, e, e − 1)  18  B_{e}  −−
(e + 1, e + 1, e + 1, e − 1)  18  B_{e−1}  −−
(2e + 1, 2e − 1)  (e + 3, e + 1, e, e − 2)  16  B_{e}  −−
(e + 2, e + 1, e, e − 1)  16  B_{e}  −−
(e + 1, e + 1, e, e)  17  B_{e}  −−
(2e, 2e)  (e + 2, e + 2, e, e − 2)  17  B_{e−2}  −−
(e + 2, e + 1, e, e − 1)  16  B_{e−1}  A_{1}
(e + 1, e + 1, e + 1, e − 1)  17  B_{e−1}  −−
(e + 2, e, e, e)  15  ∅  −−
(e + 1, e + 1, e, e)  17  ∅  −−
(2e + 2, 2e − 1)  (e + 4, e + 1, e, e − 2)  16  B_{e}  A_{1}
(e + 3, e + 1, e, e − 1)  16  B_{e}  A_{1}
(e + 2, e + 1, e, e)  17  B_{e}  A_{1}
(e + 1, e + 1, e + 1, e)  17  B_{e}  A_{1}
(2e + 1, 2e)  (e + 3, e + 2, e, e − 2)  17  B_{e}  −−
(e + 3, e + 1, e, e − 1)  15  B_{e}  A_{2}
(e + 2, e + 2, e, e − 1)  16  B_{e}  A_{1}
(e + 2, e + 1, e + 1, e − 1)  17  B_{e−1}  −−
(e + 2, e + 1, e, e)  16  B_{e}  −−
(e + 1, e + 1, e + 1, e)  18  B_{e}  −−

S_{e+1,e+1,e+1,e−1} which do not deform into each other but are both deformations of S_{e+2,e+1,e,e−1} and both deform to S_{e+1,e+1,e,e}. Furthermore we give the number of moduli for each family, in the column M. In the table we also list the base locus of |2H − b_{1}R| (which contains that of |2H − b_{2}R|). The base locus is a subscroll, for which we use the following notation [Reid 1997, 2.8]: we denote by B_{a} the subscroll corresponding to the subset of all e_{i} with e_{i} ≤ a, defined by the equations z^{(j)} = 0 for e_{j} > a. We give the number and type of the singularities of the general element; the number given in the second half of [Brawner 1997, Table A.2] is not correct. As example of the computations we look at (e + 3, e, e − 1, e − 2) with (b_{1}, b_{2}) = (2e, 2e − 2). The two equations have the form

p_{1}xw + p_{2}xz + p_{0}y^{2} + p_{3}yx + p_{6}x^{2}
q_{0}z^{2} + q_{0}yw + q_{3}xw + q_{1}yz + q_{4}xz + q_{2}y^{2} + q_{5}yx + q_{8}x^{2} ,

where the index denotes the degree in (s : t). We first use coordinate transformations to simplify these equations. By replacing y, z and w by suitable multiples we may assume that the three constant polynomials are 1. Now replacing z by z − \frac{1}{2}q_{1}y − \frac{1}{2}q_{4}x removes the yz and xz terms. We then replace y by y − q_{3}x to get rid of the xw term. By changing w we finally achieve the form z^{2} + yw + q_{8}x^{2}. By a change in (s: t) we may assume that p_{1} = s. We now look at the affine chart (w = 1, t = 1) and find y = −z^{2} − q_{8}x^{2}, which we insert in the other equation to get an equation of the form x(s+p_{2}z +⋯)+z^{4}, which is an A_{3}. We leave it again to the reader to analyze which further singularities can occur if e_{4} = 0.

2. Infinitesimal deformations. Deformations of cones over complete intersections on scrolls need not preserve the rolling factors format. We shall study in detail those who do. Many deformations of negative weight are of this type.

Definition 2.1. A pure rolling factors deformation is a deformation in which the scroll is undeformed and only the equations on the scroll are perturbed.

This means that the deformation of the additional equations can be written with the rolling factors. Such deformations are always unobstructed. However this is not the only type of deformation for which the scroll is not changed. In weight zero one can have deformations inside the scroll, where the type (b_{1}, …, b_{l}) changes.

Definition 2.2. A (general) rolling factors deformation is a deformation in which the scroll is deformed and the additional equations are written in rolling factors with respect to the deformed scroll.

The equations for the total space of a 1-parameter rolling factors deformation describe a scroll of one dimension higher, containing a subvariety of the same codimension, again in rolling factors format. Deformations over higher dimensional base spaces may be obstructed. Again in weight zero one can have deformations of the scroll, where also the type (b_{1}, …, b_{l}) changes. Finally there are non-scrollar deformations, where the perturbation of the scrollar equations does not define a deformation of the scroll. Examples of this phenomenon are easy to find (but difficult to describe explicitly). A trigonal canonical curve is a divisor in a scroll, whereas the general canonical curve of the same genus g is not of this type: the codimension of the trigonal locus in moduli space is g − 4.

Example 2.3. To give an example of a deformation inside a scroll, we let C be a tetragonal curve in ℙ^{9} with invariants (2, 2, 2; 3, 1). Then there is a weight 0 deformation to a curve of type (2, 2, 2; 2, 2). To be specific, let C be given by P = sx^{2} + ty^{2} + (s + t)z^{2}, Q = t^{3}x^{2} + s^{3}y^{2} + (s^{3} − t^{3})z^{2}. We do not deform the scroll, but only the additional equations:

x_{0}^{2} + y_{0}y_{1} + z_{0}^{2} + ε(z_{1}^{2} − x_{1}^{2})
x_{0}x_{1} + y_{1}^{2} + z_{0}z_{1} + ε(z_{1}z_{2} − x_{1}x_{2})
x_{1}^{2} + y_{1}y_{2} + z_{1}^{2} + ε(y_{0}y_{1} + z_{0}z_{1})
x_{1}x_{2} + y_{2}^{2} + z_{1}z_{2} + ε(y_{1}^{2} + z_{1}^{2})
x_{1}x_{2} + y_{0}^{2} + z_{0}^{2} − z_{1}z_{2}
x_{2}^{2} + y_{0}y_{1} + z_{0}z_{1} − z_{2}^{2}

For ε ≠ 0 we can write the ideal as

\begin{matrix}x_{0}^{2} + y_{0}y_{1} + z_{0}^{2} & + ε(z_{1}^{2} − x_{1}^{2}) \\ x_{0}x_{1} + y_{1}^{2} + z_{0}z_{1} & + ε(z_{1}z_{2} − x_{1}x_{2}) \\ x_{1}^{2} + y_{1}y_{2} + z_{1}^{2} & + ε(z_{2}^{2} − x_{2}^{2}) \\ x_{0}x_{1} + y_{1}^{2} + z_{0}z_{1} & + ε(y_{0}^{2} + z_{0}^{2}) \\ x_{1}^{2} + y_{1}y_{2} + z_{1}^{2} & + ε(y_{0}y_{1} + z_{0}z_{1}) \\ x_{1}x_{2} + y_{2}^{2} + z_{1}z_{2} & + ε(y_{1}^{2} + z_{1}^{2}) \end{matrix}

We can describe this deformation in the following way. Write Q = sQ_{s} + tQ_{t}. The two times three equations above are obtained by rolling factors from sP − εQ_{t} and tP + εQ_{s}. We may generalize this example.

Lemma 2.4. Let V be a complete intersection of divisors of type aH − b_{1}R, aH − b_{2}R, given by equations P, Q. If it is possible to write Q = sQ_{s} + t^{b_{1}−b_{2}−1}Q_{t} then the equations sP − εQ_{t}, t^{b_{1}−b_{2}−1}P + εQ_{s} give a deformation to a complete intersection of type aH − (b_{1} − 1)R, aH − (b_{2} + 1)R.

In general one has to combine such a deformation with a deformation of the scroll.

(2.5) Non-scrollar deformations.

Example 2.6. As mentioned before such deformations must exist in weight zero for trigonal cones. We proceed with the explicit computation of embedded deformations. We start from the normal bundle exact sequence

0 ⟶ N_{S/C} ⟶ N_{C} ⟶ N_{S} ⊗ O_{C} ⟶ 0 .

As C is a curve of type 3H − (g − 4)R on S we have that C · C = 3g + 6 and H^{1}(C, N_{S/C}) = 0. So we are interested in H^{0}(C, N_{S} ⊗ O_{C}), and more particularly in the cokernel of the map H^{0}(S, N_{S}) ⟶ H^{0}(C, N_{S} ⊗ O_{C}), as H^{0}(S, N_{S}) gives deformations of the scroll.

Proposition 2.7. The cokernel of the map H^{0}(S, N_{S}) ⟶ H^{0}(C, N_{S} ⊗ O_{C}) has dimension g − 4.

Proof . An element of H^{0}(C, N_{S} ⊗ O_{C}) is a function φ on the equations of the scroll such that the generators of the module of relations map to zero in O_{C} and it lies in the image of H^{0}(S, N_{S}) if the function values can be lifted to O_{S} such that the relations map to 0 ∈ O_{S}. Therefore we perform our computations in O_{S}. We have to introduce some more notation. Using the equations described in (1.3) we have three types of scrollar equations, f_{i,j} = x_{i}x_{j+1} − x_{i+1}x_{j}, g_{i,j} = y_{i}y_{j+1} −y_{i+1}y_{j} and mixed equations h_{i,j} = x_{i}y_{j+1} −x_{i+1}y_{j}. The scrollar relations come from doubling a row in the matrix and there are two ways to do this. The equations resulting from doubling the top row can be divided by s, and the other ones by t, so the result is the same. A relation involving only equations of type f_{i,j} gives the condition

xs^{e_{1}−i−1}t^{i}φ(f_{j,k}) − xs^{e_{1}−j−1}t^{j}φ(f_{i,k}) + xs^{e_{1}−k−1}t^{k}φ(f_{i,j}) = 0 ∈ O_{C}

which may be divided by x. As the image φ(f_{i,j}) is quadratic in x and y the resulting left hand side cannot be a multiple of the equation of C, so we have

s^{e_{1}−i−1}t^{i}φ(f_{j,k}) − s^{e−1−j−1}t^{j}φ(f_{i,k}) + s^{e_{1}−k−1}t^{k}φ(f_{i,j}) = 0 ∈ O_{S}

and the analogous equation involving only the g_{i,j} equations. For the mixed equations we get

xs^{e_{1}−i−1}t^{i}φ(h_{j,k}) − xs^{e_{1}−j−1}t^{j}φ(h_{i,k}) + ys^{e_{2}−k−1}t^{k}φ(f_{i,j}) = ψ_{i,j;k}P ∈ O_{S}

with ψ_{i,j;k} of degree e_{1} + e_{2} − 3 = g − 5 and analogous ones involving g_{i,j} with coefficients ψ_{i;j,k}. These coefficients are not independent, but satisfy a systems of equations coming from the syzygies between the relations. They can also be verified directly. We obtain

s^{e_{1}−i−1}t^{i}ψ_{j,k;l} − s^{e_{1}−j−1}t^{j}ψ_{i,k;l} + s^{e_{1}−k−1}t^{k}ψ_{i,j;l} = 0 ∈ O_{S}

and

xs^{e_{1}−i−1}t^{i}ψ_{j;k,l} − xs^{e_{1}−j−1}t^{j}ψ_{i;k,l} + ys^{e_{2}−k−1}t^{k}ψ_{i,j;l} − ys^{e_{2}−l−1}t^{l}ψ_{i,j;k} = 0

The last set of equations shows that s^{e_{2}−k−1}t^{k}ψ_{i,j;l} = s^{e_{2}−l−1}t^{l}ψ_{i,j;k} (rolling factors!) and therefore ψ_{i,j;k} = s^{e_{2}−k−1}t^{k}ψ_{i,j;} with ψ_{i,j;} of degree e_{1} − 2. This yields the equations

s^{e_{1}−i−1}t^{i}ψ_{j,k;} − s^{e_{1}−j−1}t^{j}ψ_{i,k;} + s^{e_{1}−k−1}t^{k}ψ_{i,j;} = 0

Our next goal is to express all ψ_{i,j;} in terms of the ψ_{i,i+1;} (where 0 ≤ i ≤ e_{1} − 2). First we observe by using the last equation for the triples (0, i, i + 1) and (i, i + 1, e_{1} − 1) that ψ_{i,i+1;} is divisible by t^{i} and by s^{e_{1}−i−2} so ψ_{i,i+1;} = s^{e_{1}−i−2}t^{i}c_{i} for some constant c_{i}. By induction it then follows that ψ_{i,j;} = s^{e_{1}−i−2}t^{i}c_{j−1} + s^{e_{1}−i−3}t^{i+1}c_{j−2} + ⋯ + s^{e_{1}−j−1}t^{j−1}c_{i}, so the solution of the equations depends on e_{1} − 1 constants. Similarly one finds e_{2} − 1 constants d_{i} for the ψ_{i;j,k} so altogether e_{1} + e_{2} − 2 = g − 4 constants. Finally we can solve for the perturbations of the equations. We give the formulas in the case that all d_{i} and all c_{i} but one are zero, say c_{γ} = 1. This implies that ψ_{i,j;} = 0 if γ ∉ [i, j) and ψ_{i,j;} = s^{e_{1}−i−j+γ−1}t^{i+j−γ−1} if γ ∈ [i, j); under the last assumption ψ_{i,j;k} = s^{g−4−i−j−k+γ}t^{i+j+k−γ−1}. We take φ(f_{i,j}) = 0 if γ ∉ [i, j). It follows that for a fixed k the φ(h_{i,k}) with i ≤ γ are related by rolling factors, as are the φ(h_{i,k}) with i > γ. This reduces the mixed equations with fixed k to one, which can be solved for in a uniform way for all k. To this end we write the equation P as

(s^{2e_{1}−e_{2}−γ+2}A_{γ}^{+} + t^{γ+1}A_{2e_{1}−e_{2}−γ+1}^{−})x^{3} + y(B_{e_{1}+2}x^{2} + C_{e_{2}+2}xy + D_{2e_{2}−e_{1}+2}y^{2})

which we will abbreviate as (s^{2e_{1}−e_{2}−γ+2}A^{+} + t^{γ+1}A^{−})x^{3} + yE. We set

φ(f_{i,j}) = 0,  if γ ∉ [i, j)
φ(f_{i,j}) = s^{e_{1}−i−j−1+γ}t^{i+j−γ−1}E,  if γ ∈ [i, j)
φ(g_{i,j}) = 0,
φ(h_{i,k}) = −s^{e_{2}−1−k−i+γ}t^{i+k}A^{−}x^{2}, if i ≤ γ
φ(h_{i,k}) = s^{2e_{1}+1−i−k}t^{i+k−γ−1}A^{+}x^{2}, if i > γ

This is well defined, because all exponents of s and t are positive. □

A similar computation can be used to show that all elements of T^{1}(ν) with ν > 0 can be written rolling factors type. However, even more is true, they can be represented as pure rolling factors deformations, see [Drewes–Stevens 1996], where a direct argument is given.

We generalize the above discussion to the case of a complete intersection of divisors of type aH − b_{i}R (with the same a ≥ 2) on a scroll

( \begin{matrix}z0^{(1)} & … & zd_{1}−1^{(1)} & … & z0^{(k)} & … & zd_{k}−1^{(k)} \\ z1^{(1)} & … & zd_{1}^{(1)} & … & z1^{(k)} & … & zd_{k}^{(k)} \end{matrix} ) .

We have equations f_{ij}^{(αβ)} = zi^{(α)}zj+1^{(β)} − zi+1^{(α)}zj^{(β)}. The lowest degree in which non rolling factors deformations can occur is a − 3. We get the conditions

z^{(α)}s^{d_{α}−i−1}t^{i}φ(f_{jk}^{(βγ)}) − z^{(β)}s^{d_{β}−j−1}t^{j}φ(f_{ik}^{(αγ)}) + z^{(γ)}s^{d_{γ}−k−1}t^{k}φ(f_{ij}^{(αβ)})

= ∑_{l} ψ_{ijk;l}^{(αβγ)}P^{(l)}

with the ψ_{ijk;l}^{(αβγ)} homogeneous polynomials in (s:t) of degree b_{l}−1. The relations between these polynomials come from the syzygies of the scroll: we add four of these relations, multiplied with a term linear in the z^{(α)}; then the left hand side becomes zero, leading to a relation (in O_{S}) between the P^{(l)}. As we are dealing with a complete intersection, the relations are generated by Koszul relations. Because the coefficients of the relation obtained are linear in the z^{(α)}, they cannot lie in the ideal generated by the P^{(l)} (as a ≥ 2), so they vanish and we obtain for each l equations

z^{(α)}s^{d_{α}−i−1}t^{i}ψ_{jkm;l}^{(βγδ)} − z^{(β)}s^{d_{β}−j−1}t^{j}ψ_{ikm;l}^{(αγδ)}
+ z^{(γ)}s^{d_{γ}−k−1}t^{k}ψ_{ijm;l}^{(αβδ)} − z^{(δ)}s^{d_{δ}−m−1}t^{m}ψ_{ijk;l}^{(αβγ)} = 0 .

Here some of the α, …, δ may coincide. If e.g. δ is different from α, β and γ, then ψ_{ijk;l}^{(αβγ)} = 0. If there are at least four different indices (e.g. if the scroll is nonsingular of dimension at least four) then δ can always be chosen in this way, so all coefficients vanish and every deformation of degree a−3 is of rolling factors type. Suppose now the scroll is a cone over a nonsingular 3-dimensional scroll, i.e. we have three different indices at our disposal. Then every ψ_{ijk;l}^{(αβγ)} with at most two different upper indices vanishes, and the ones with three different indices satisfy rolling factors equations. We conclude that for pairwise different α, β, γ

ψ_{ijk;l}^{(αβγ)} = s^{d−i−j−k−3}t^{i+j+k}ψ′_{l}

with d = d_{α} + d_{β} + d_{γ} the degree of the scroll. Finally, for the cone over a 2-dimensional scroll we get similar computations as in the trigonal example above.

Proposition 2.8. A tetragonal cone (with g > 5) has non-scrollar deformations of degree −1 if and only if b_{2} = 0. If the canonical curve lies on a Del Pezzo surface then the dimension is 1. If the curve is bielliptic then the dimension is b_{1} = g − 5.

Proof . First suppose e_{3} > 0. Then the only possibly non zero coefficients are the ψ′_{l}, which have degree b_{l} + 2 − ∑ e_{i}. As b_{1} + b_{2} = ∑ e_{i} − 2 they do not vanish iff b_{2} = 0. In this case the computation yields one non rolling factors deformation of the Del Pezzo surface on which the curve lies.

If e_{3} = 0, then b_{2} = 0. For a bielliptic curve the methods above yield (e_{1} − 1) + (e_{2} − 1) = b_{1} = g − 5 non-scrollar deformations (a detailed computation is given in [Brawner 1996]). Suppose now that the curve lies on a (singular) Del Pezzo surface. If b_{1} = e_{1} > e_{2} = 2 then the equation P contains the monomial xz with nonzero coefficient, which we take to be 1, while there is no monomial yz. After a coordinate transformation we may assume that the same holds in case e_{1} = e_{2} = 2. Let φ(h_{i,k}) ≡ ζ_{i,k}z mod (x, y). In the equation

xs^{e_{1}−i−1}t^{i}φ(g_{j,k}) − ys^{e_{2}−j−1}t^{j}φ(h_{i,k}) + ys^{e_{2}−k−1}t^{k}φ(h_{i,j}) = ψ_{i;j,k}P

holding in O_{S} the monomial yz occurs only on the left hand side, which shows that the ζ_{i,k} are of rolling factors type in the first index. Being constants, they vanish. This means that in the equation

xs^{e_{1}−i−1}t^{i}φ(h_{j,k}) − xs^{e_{1}−j−1}t^{j}φ(h_{i,k}) + ys^{e_{2}−k−1}t^{k}φ(f_{i,j}) = ψ_{i,j;k}P

the monomial xz does not occur on the left hand side and therefore ψ_{i,j;k} = 0. We find only e_{2} − 1 = 1 non rolling factors deformation. If e_{1} = 2, e_{2} = 1 we find one deformation. Finally, if e_{1} = 3, e_{2} = 1 then there is only one type of mixed equation. We have two constants c_{0} and c_{1}. Let the coefficient of xz in P be p_{0}s + p_{1}t. We obtain the equations

sζ_{1,0} − tζ_{0,0} = c_{0}(p_{0}s + p_{1}t)
sζ_{2,0} − tζ_{1,0} = c_{1}(p_{0}s + p_{1}t)

from which we conclude that p_{0}c_{0} + p_{1}c_{1} = 0, giving again only one non rolling factors deformation. □

(2.9) Rolling factors deformations of degree −1. We look at the miniversal deformation of the scroll:

( \begin{matrix}z0^{(1)} & … & zd_{1}−2^{(1)} & zd_{1}−1^{(1)} & z0^{(2)} & … & zd_{k}−2^{(k)} & zd_{k}−1^{(k)} \\ z1^{(1)} + ζ1^{(1)} & … & zd_{1}−1^{(1)} + ζd_{1}−1^{(1)} & zd_{1}^{(1)} & z1^{(2)} & … & zd_{k}−1^{(k)} + ζd_{k}−1^{(k)} & zd_{k}^{(k)} \end{matrix} )

To compute which of those deformations can be lifted to deformations of a complete intersection on the scroll we have to compute perturbations of the additional equations. We assume that we have a complete intersection of divisors of type aH − b_{i}R (with the same a ≥ 2). Extending the notation introduced before we write the columns in the matrix symbolically as (z_{α}, z_{α+1}+ζ_{α+1}). In order that this makes sense for all columns we introduce dummy variables ζ0^{(i)} and ζd_{i}^{(i)} with the value 0. The Koszul type relations give no new conditions, but the relation

P_{m+1}z_{β} − P_{m}z_{β+1} − ∑_{α} p_{α,m}f_{βα} = 0

gives as equation in the local ring for the perturbations P′_{m} of P_{m}:

P′_{m+1}z_{β} − P′_{m}z_{β+1} − ∑_{α} p_{α,m}(ζ_{α+1}z_{β} − z_{α}ζ_{β+1}) = 0 .

In particular we see that we can look at one equation on the scroll at a time. As ∑ p_{α,m}z_{α} = P_{m} the coefficient of ζ_{β+1} vanishes. Because tz_{β} − sz_{β+1} = 0 we get a condition which is independent of β:

sP′_{m+1} − tP′_{m} − s ∑_{α} p_{α,m}ζ_{α+1} = 0

This has to hold in the local ring, but as the degree of the p_{α,m} is lower than that of the equations defining the complete intersection on the scroll (here we use the assumption that all degrees a are equal), it holds on the scroll. From it we derive the equation

s^{b}P′_{b} − t^{b}P′_{0} = ∑_{m=0}^{b−1} ∑_{α} s^{m+1}t^{b−m−1}p_{α,m}ζ_{α+1}  (S)

which has to be solved with P′_{b} and P′_{0} polynomials in the z_{α} of degree a − 1. We determine the monomials on the right hand side. The result depends on the chosen equations, but only on P_{0} and P_{b} and not on the intermediate ones, provided they are obtained by rolling factors.

Example 2.10. Let b = 4. We take variables y_{i} = s^{3−i}t^{i}y, z_{i} = s^{3−i}t^{i}z with deformations η_{i}, ζ_{i}, and roll from y_{0}z_{0} to y_{2}z_{2} in two different ways:

y_{0}z_{0} → y_{1}z_{0} → y_{1}z_{1} → y_{2}z_{1} → y_{2}z_{2}
y_{0}z_{0} → y_{0}z_{1} → y_{0}z_{2} → y_{1}z_{2} → y_{2}z_{2}

This gives as right-hand side of the equation (S) in the two cases

s^{4}t^{3}zη_{1} + s^{4}t^{3}yζ_{1} + s^{5}t^{2}zη_{2} + s^{5}t^{2}yζ_{2}
s^{4}t^{3}yζ_{1} + s^{5}t^{2}yζ_{2} + s^{4}t^{3}zη_{1} + s^{5}t^{2}zη_{2}

which is the same expression. Similarly, if we roll from z_{0}^{2} to z_{2}^{2} we get

2s^{4}t^{3}zζ_{1} + 2s^{5}t^{2}zζ_{2}

However, if we roll in the last step from y_{1}z_{2} to y_{1}z_{3} we get

s^{4}t^{3}yζ_{1} + s^{5}t^{2}yζ_{2} + s^{4}t^{3}zη_{1}

(remember that we have no deformation parameter ζ_{3}).

To analyze the general situation it is convenient to use multi-index notation. The equation P of a divisor in |aH − bR| may then be written as

P = ∑_{|I|=a} ∑_{j=0}^{⟨e,I⟩−b} p_{I,j}s^{⟨e,I⟩−b−j}t^{j}z^{I} .

Here e = (e_{1}, …, e_{k}) is the vector of degrees and z^{I} stands for (z^{(1)})^{i_{1}} · . . . · (z^{(k)})^{i_{k}}.

Proposition 2.11. The lifting condition for the equations P_{m} is that for each I with |I| = a − 1 and ⟨e, I⟩ < b − 1 the following b − ⟨e, I⟩ − 1 linear equations hold:

∑_{l=1}^{k} ∑_{j=0}^{⟨e,I+δ_{l}⟩−b}(i_{l} + 1)p_{I+δ_{l},j}ζj+n^{(l)} = 0 ,

where 0 < n < b − ⟨e, I⟩

Proof . We look at a monomial s^{⟨e,I′⟩−b−j}t^{j}z^{I′}. In rolling from P_{0} to P_{m} we go from z_{A} to z_{A+B}. Here we write a monomial as product of a factors: z_{α_{1}} ⋯ z_{α_{a}} with i′_{l} factors of type l. Let I′ = I + δ_{l} with δ_{l} the lth unit vector. The monomial leads to an expression in which the coefficient of z^{I} is

∑_{{q|α_{q} of type l}} ∑_{r=1}^{β_{q}} s^{⟨e,I⟩+r−j+α_{q}}t^{b−r+j−α_{q}}ζ_{α_{q}+r}

We stress that the choice of α_{q} can be very different for different j. We collect all contributions and look at the coefficient of s^{⟨e,I⟩+n}t^{b−n}z^{I} with 0 < n < b − ⟨e, I⟩. This cannot be realized as left-hand side of equation (S). Because b − n = b − r + j − α_{q} this coefficient is

∑_{l=1}^{k} ∑_{j=0}^{⟨e,I+δ_{l}⟩−b}(i_{l} + 1)p_{I+δ_{l},j}ζj+n^{(l)} = 0 ,

We note that all terms really occur: in rolling from z_{A} to z_{A+B} we have to increase the qth factor sufficiently many times, because ⟨e, I⟩ < b − 1. □

Example 2.12: trigonal cones. Let the curve be given by the bihomogeneous equation

F = A_{2a−m+2}z^{3} + B_{a+2}z^{2}w + C_{m+2}zw^{2} + D_{2m−a+2}w^{3}

then there are only conditions for I = (0, 2), i.e. for w^{2}, as a+m = g−2 > b−1. So if 2m < b − 1 = g − 5 we get b − 2m − 1 = a − m − 3 equations on the deformation variables ζ_{1}, …, ζ_{a−1}, ω_{1}, …, ω_{m−1}

∑_{j=0}^{m+2} c_{j}ζ_{j+n} + 3 ∑_{j=0}^{2m−a+2} d_{j}ω_{j+n} = 0 ,

as stated in [Drewes–Stevens 1996, 3.11]. We have a system of linear equations so we can write the coefficient matrix. It consists of two blocks ( C | D) with C of the form

( \begin{matrix}c_{0} & c_{1} & c_{2} & … & c_{m+2} & 0 & 0 & … & 0 & 0 & 0 \\ 0 & c_{0} & c_{1} & … & c_{m+1} & c_{m+2} & 0 & … & 0 & 0 & 0 \\ 0 & 0 & c_{0} & … & c_{m} & c_{m+1} & c_{m+2} & … & 0 & 0 & 0 \\ & & & ... & & & & ... \\ 0 & 0 & 0 & … & c_{0} & c_{1} & c_{2} & … & c_{m+2} & 0 & 0 \\ 0 & 0 & 0 & … & 0 & c_{0} & c_{1} & … & c_{m+1} & c_{m+2} & 0 \\ 0 & 0 & 0 & … & 0 & 0 & c_{0} & … & c_{m} & c_{m+1} & c_{m+2} \end{matrix} )

and D similarly. Obviously this system has maximal rank.

The Proposition gives a system of linear equations and we call the coefficient matrix lifting matrix. It was introduced for tetragonal cones in [Brawner 1996]. In general the lifting matrix will have maximal rank, but it is a difficult question to decide when this happens.

Example 2.13: trigonal K3s. We take the invariants (e, e, e), b = 3e − 2 with e ≥ 3. The K3 lies on ℙ^{2} × ℙ^{1} and is given by an equation of bidegree (3, 2). Now there are six Is with |I| = 2 each giving rise to e − 3 equations in 3(e − 1) deformation variables. In general the matrix has maximal rank, but for special surfaces the rank can drop. Consider an equation of type p_{1}x^{3}+p_{2}y^{3}+p_{3}z^{3} with the p_{i} quadratic polynomials in (s:t) without common or multiple zeroes. Then the surface is smooth. The lifting equations corresponding to the quadratic monomials xy, xz and yz vanish identically and the lifting matrix reduces to a block-diagonal matrix of rank 3(e − 3). The kernel has dimension 6, but the corresponding deformations are obstructed: an extension of the K3 would be a Fano 3-fold with isolated singularities lying as divisor of type 3H − bR on a scroll S(e_{1}, e_{2}, e_{3}, e_{4}), and a computation reveals that such a Fano can only exist for ∑ e_{i} ≤ 8.

We can say something more for the lifting conditions coming from one quadratic equation.

Proposition 2.14. The lifting matrix for one quadratic equation has dependent rows if and only if the generic fibre has a singular point on the subscroll B_{b−1}.

Proof . The equation P on the scroll can be written in the form ^{t}zΠz with Π a symmetric k × k matrix with polynomials in (s : t) as entries. The condition that there is a singular section of the form z = (0, …, 0, z^{(l+1)}(s, t), …, z^{(k)}(s, t)) with e_{l} > b − 1 is that ^{t}zΠ = 0 or ^{t}z_{>l}Π_{>l} = 0 where z_{>l} = (z^{(l+1)}(s, t), …, z^{(k)}(s, t)) and Π_{>l} is the matrix consisting of the last k − l rows of Π. The resulting system of equations for the coefficients of the polynomials z^{(i)}(s, t) gives exactly the lifting matrix. □

(2.15) Tetragonal curves. Most of the following results are contained in the preprint [Brawner 1996]. We have two equations on the scroll and the lifting matrix M can have rows coming from both equations. We first suppose that b_{2} > 0. Then also e_{3} > 0 and the number of columns of M is always ∑(e_{i}−1) = g − 6, but the number of rows depends on the values of (e_{1}, e_{2}, e_{3}; b_{1}, b_{2}): it is ∑_{i,j} max(0, b_{i} − e_{j} − 1).

Theorem 2.16. Let X be the cone over a tetragonal canonical curve and suppose that b_{2} > 0. Then dim T_{X}^{1}(−2) = 0. Suppose that the g_{4}^{1} is not composed with an involution of genus \frac{b_{2}}{2} + 1.

1) If b_{1} < e_{1} + 1 or b_{2} < e_{3} + 1 or g ≤ 15 then dim T_{X}^{1}(−1) = 9 + dim Cork M.

2) If b_{1} ≥ e_{1} +1, b_{2} ≥ e_{3} +1 and g > 15 then 9+dim Cork M ≤ dim T_{X}^{1}(−1) ≤ \frac{g+3}{6} +6+dim Cork M and the maximum is obtained for g of the form 6n−3 and (e_{1}, e_{2}, e_{3}; b_{1}, b_{2}) = (3n − 2, 2n − 2, n − 2; 4n − 4, 2n − 4).

3) For generic values of the moduli dim Cork M = 0.

Proof . If b_{2} > 0 there are only rolling factors deformations in negative degrees. In particular dim T_{X}^{1}(−2) = 0. The number of pure rolling factors deformations is ρ = ∑_{i,j} max(e_{j} − b_{i} + 1, 0). The number of rows in the lifting matrix is ∑_{i,j} max(0, b_{i} − e_{j} − 1) = 3(b_{1} + b_{2}) − 2(e_{1} + e_{2} + e_{3} + 3) + ρ = g − 15 + ρ. If ρ > 9 the number of rows exceeds the number of columns and dim T_{X}^{1}(−1) = ρ + dim Cork M; otherwise it is 9 + dim Cork M. So we have to estimate ρ. As the g_{4}^{1} is not composed we have b_{1} ≤ e_{1} + e_{3}. Together with b_{1} ≤ 2e_{2} we get 3b_{1} ≤ 2g − 6 and 3b_{2} ≥ g − 9; from e_{1} ≤ \frac{g−1}{2} we now derive e_{1} − b_{2} + 1 ≤ \frac{g−1}{2} + 1 − \frac{g−9}{3}. Also b_{2} = e_{1} + e_{2} + e_{3} − 2 − b_{1} ≥ e_{2} − 2, so e_{2} − b_{2} + 1 ≤ 3.

2) Suppose first that b_{1} ≥ e_{1} + 1 and b_{2} ≥ e_{3} + 1. Then ρ = max(0, e_{1} − b_{2} + 1) + max(0, e_{2} − b_{2} + 1) ≤ \frac{g+3}{6} + 6. Equality is achieved iff e_{1} = (g − 1)/2, b_{2} = (g − 9)/3 and e_{2} = b_{2} + 2, so g has the form 6n − 3 and (e_{1}, e_{2}, e_{3}; b_{1}, b_{2}) = (3n − 2, 2n − 2, n − 2; 4n − 4, 2n − 4).

1) In all other cases ρ ≤ 9: if b_{1} ≥ e_{1} + 1, but b_{2} < e_{3} + 1 then ρ = (e_{1} − b_{2} + 1) + (e_{2} − b_{2} + 1) + (e_{3} − b_{2} + 1) = g − 3b_{2} ≤ 9. If e_{2} + 1 ≤ b_{1} < e_{1} + 1 then ρ = (e_{1} − b_{1} + 1) + (e_{1} − b_{2} + 1) + max(0, e_{2} − b_{2} + 1) + max(0, e_{3} − b_{2} + 1) = 2e_{1} + 7 − g + max(0, e_{2} − b_{2} + 1) + max(0, e_{3} − b_{2} + 1). As b_{2} > 0 we have that max(0, e_{2} − b_{2} + 1) + max(0, e_{3} − b_{2} + 1) > max(0, e_{2} + e_{3} − b_{2} + 1). But from b_{1} ≤ e_{1} it follows that b_{2} ≥ e_{2} + e_{3} − 2. So ρ ≤ (g − 1) + 7 − g + 3 = 9. If b_{1} < e_{2} + 1 then ρ ≤ 2e_{1} + 2e_{2} + 4 − 2b_{1} − 2b_{2} + 2e_{3} = 8.

3) It is easy to construct lifting matrices of maximal rank for all possible numbers of blocks occurring. □

Now we consider the case that the g_{4}^{1} is composed with an involution of genus g′ = \frac{b_{2}}{2} + 1. So if b_{2} > 0, then g′ > 1. After a coordinate transformation we may assume that the surface Y is singular along the section x = y = 0, so its equation depends only on x and y: P = P (x, y; s, t). We may assume that Q has the form Q = z^{2} + Q′(x, y; s, t). Let M_{xy} be the submatrix of the lifting matrix consisting of the blocks coming from P and Q′ and the ξ and η deformations.

Theorem 2.17. Let X be a tetragonal canonical cone such that the g_{4}^{1} is composed with an involution of genus g′ > 1. Then dim T_{X}^{1}(−1) = e_{1} + e_{2} − 2e_{3} + 6 + Cork M_{xy}.

Proof . The rows in the lifting matrix M coming from the first equation and the variable z vanish identically. The second equation gives a z-block which is an identity matrix of size b_{2} − e_{3} − 1 = e_{3} − 1, so all ζ variables have to vanish. What remains is the matrix M_{xy} which has e_{1} + e_{2} − 2 columns. The number of rows is max(0, e_{2} − e_{3} − 3) + max(0, e_{1} − e_{3} − 3) + max(0, 2e_{3} − e_{1} − 1) + max(0, 2e_{3} − e_{2} − 1). We estimate the last two terms with e_{3} − 1 and the first two by e_{2} − e_{3}, resp. e_{1} − e_{3}. Therefore the number of rows is at most e_{1} + e_{2} − 2. For each term which contributes 0 to the sum we have pure rolling factors deformations, so if the matrix has maximal rank the dimension of T_{X}^{1}(−1) is e_{1} + e_{2} − 2 − (2e_{3} − 8). □

Example 2.18. It is possible that the lifting matrix M does not have full rank even if the g_{4}^{1} is not composed. An example with invariants (6, 5, 5; 7, 7) is the curve given by the equations (s^{5} +t^{5})x^{2} +s^{3}y^{2} +t^{3}z^{2}, s^{5}x^{2} +(s^{3} −t^{3})(y −z)^{2} + 2t^{3}z^{2}. The matrix is

( \begin{matrix}0 & … & 0 & | & 2 & 0 & 0 & 0 & | & 0 & 0 & 0 & 0 \\ 0 & … & 0 & | & 0 & 0 & 0 & 0 & | & 0 & 0 & 0 & 2 \\ 0 & … & 0 & | & 2 & 0 & 0 & −2 & | & −2 & 0 & 0 & 2 \\ 0 & … & 0 & | & −2 & 0 & 0 & 2 & | & 2 & 0 & 0 & 2 \end{matrix} ) .

Finally we mention the case b_{2} = 0. Bielliptic curves (e_{3} = 0) are treated in [Ciliberto–Miranda 1992], curves on a Del Pezzo in [Brawner 1996] (but he overlooks those with e_{3} = 0). Now there is only one equation coming from Q, which can be perturbed arbitrarily. As the z variable does not enter the scroll, we have one coordinate transformation left. The lifting matrix involves only rows coming from the equation P. One checks that the matrix M resp. M_{xy} has maximal rank and the number of rows does not exceed the number of columns. Together with the number of non-scrollar deformations (Prop. 2.8) this yields the following result, where we have excluded the complete intersection case g = 5.

Proposition 2.19. Let X be the cone over a tetragonal canonical curve C with b_{2} = 0 and g > 5. Then dim T_{X}^{1}(−2) = 1.

1) If C lies on a Del Pezzo surface then dim T_{X}^{1}(−1) = 10.

2) If C is bielliptic (e_{3} = 0), then dim T_{X}^{1}(−1) = 2g − 2.

Remark 2.20. For all non-hyperelliptic canonical cones the dimension of T_{X}^{1}(ν) with ν ≥ 0 is the same. The Wahl map easily gives dim T_{X}^{1}(0) = 3g − 3, dim T_{X}^{1}(1) = g, dim T_{X}^{1}(2) = 1 and dim T_{X}^{1}(ν) = 0 for ν ≥ 3 (see e.g. [Drewes– Stevens 1996], 3.3).

3. Rolling factors obstructions. Rolling factors deformations can be obstructed. We first give a general result on the dimension of T^{2}. For the case of quadratic equations on the scroll one can actually write down the base equations.

Proposition 3.1. Let X be the cone over a complete intersection of divisors of type aH − b_{i}R with b_{i} > 0 (and the same a ≥ 2) on a scroll. If a > 2, then dim T_{X}^{2}(−a) = ∑(b_{i} − 1), and dim T_{X}^{2}(−a) ≥ ∑(b_{i} − 1) in case a = 2.

Proof . Let ψ ∈ Hom(R/R_{0}, O_{X}) be an homogeneous element of degree −a. The degree of ψ(R_{α,β,γ}) is 3 − a, so ψ vanishes on the scrollar relations, if a > 2. If a = 2 we can assert that the functions vanishing on the scrollar relations span a subspace of T_{X}^{2}(−2). As the degree of the relation R_{α,m}^{n} is a + 1, the image ψ(R_{α,m}^{n}) is a linear function of the coordinates. The relations

R_{α,m}^{n}z_{β} − R_{β,m}^{n}z_{α} − ∑ R_{j,k,γ}^{n}p_{γ,m}^{n} = Pm^{(n)}f_{α,β} − f_{α,β}Pm^{(n)}.

imply that the ψ(R_{α,m}^{n}) are also in rolling factors form. A basis (of the relevant subspace) of Hom(R/R_{0}, O_{X})(−a) consists of the 2 ∑ b_{i} elements ψ_{l,s}^{i}(R_{α,m}^{j}) =

δ_{ij}δ_{lm}z_{α}, ψ_{l,t}^{i}(R_{α,m}^{j}) = δ_{ij}δ_{lm}z_{α+1}, where 0 ≤ m < b_{j}. The image of Pm^{(i)} in

Hom(R/R_{0}, O_{X})(−a) is ψ_{m−1,s}^{i} − ψ_{m,t}^{i}, if 0 < m < b_{i}, −ψ_{0,t}^{i} for m = 0, and ψ_{b_{i}−1,s}^{i} for m = b_{i}. The quotient has dimension ∑(b_{i} − 1). □

For a = 2 only the rolling factors obstructions will contribute to the base equations. A more detailed study could reveal if there are other obstructions. Typically this can happen, if there exist non-scrollar deformations. As example we mention Wahl’s result for tetragonal cones that dim T_{X}^{2}(−2) = g − 7 = b_{1} + b_{2} − 2, if b_{2} > 0, whereas for a curve on a Del Pezzo the dimension is 2(g − 6) [Wahl 1997, Thm. 5.6]. In the quadratic case we can easily write the base equations, given a first order lift of the scrollar deformations. We can consider each equation on the scroll separately, so we will suppress the upper index of the additional equations in our notation. We may assume that we have pure rolling factors deformations ρ_{α} and that the lifting conditions are satisfied. We can write the perturbation of the equation P_{m} as

P_{m}(z) + P′_{m}(z, ζ, ρ) .

Note that P′_{m} is linear in z. Now we have the following result [Stevens 1996].

Proposition 3.2. The maximal extension of the infinitesimal deformation defined by the P′_{m} is given by the b − 1 base equations

P′_{m}(ζ, ζ, ρ) − P_{m}(ζ) = 0 ,

with 1 ≤ m ≤ b − 1.

Proof . We also suppress ρ from the notation. We have to lift the relations R_{β,m}. As the lifting equations are satisfied we can write

P′_{m+1}(z, ζ)z_{β} − P′_{m}(z, ζ)z_{β+1} − ∑ p_{α,m}(z)ζ_{α+1}z_{β} = ∑ f_{βγ}d_{γ}(ζ) ,

because the left hand side lies in the ideal of the scroll. This identity involving quadratic monomials in the z-variables can be lifted to the deformation of the scroll. We write f̃_{βα} for the deformed equation z_{β}(z_{α+1} + ζ_{α+1}) − (z_{β+1} + ζ_{β+1})z_{α}. We get

P′_{m+1}(z+ζ, ζ)z_{β}−P′_{m}(z, ζ)(z_{β+1}+ζ_{β+1})−∑ p_{α,m}(z+ζ)z_{α+1}z_{β} = ∑ f̃_{βγ}d_{γ}(ζ).

We now lift the relation R_{β,m}:

(P_{m+1}(z) + P′_{m+1}(z + ζ, ζ) − P_{m+1}(ζ)) z_{β} − (P_{m}(z) + P′_{m}(z, ζ)) (z_{β+1} + ζ_{β+1})

− ∑ f̃_{βα}p_{α,m}(z) − ∑ f̃_{βγ}d_{γ}(ζ) = 0.

If 1 ≤ m ≤ b − 1, then P_{m} occurs in a relation as first and as second term. Therefore P′_{m}(z, ζ) and P′_{m}(z +ζ, ζ)−P_{m}(ζ) have to be equal. These equations correspond to the b − 1 elements of T_{X}^{2}(−2), constructed above. □

Example 3.3. We continue with our rolling factors example 2.10. We look at two ways of rolling:

y_{0}z_{0} → y_{1}z_{0} → y_{1}z_{1} → y_{2}z_{1} → y_{2}z_{2}
y_{0}z_{0} → y_{0}z_{1} → y_{0}z_{2} → y_{1}z_{2} → y_{1}z_{3}

The equation for P′_{0} and P′_{4} has a unique solution with P′_{0} = 0. We get

\begin{aligned}P′_{0} & = & 0, & 0 \\ P′_{1} & = & η_{1}z_{0}, & y_{0}ζ_{1} \\ P′_{2} & = & η_{1}z_{1} + y_{1}ζ_{1}, & y_{1}ζ_{1} + y_{0}ζ_{2} \\ P′_{3} & = & η_{1}z_{2} + y_{2}ζ_{1} + η_{2}z_{1}, & y_{2}ζ_{1} + y_{1}ζ_{2} + η_{1}z_{2} \\ P′_{4} & = & η_{1}z_{3} + y_{3}ζ_{1} + η_{2}z_{2} + y_{2}ζ_{2}, & y_{3}ζ_{1} + y_{2}ζ_{2} + η_{1}z_{3} \end{aligned}

The resulting base equations are in both cases

0, η_{1}ζ_{1}, η_{1}ζ_{2} + η_{2}ζ_{1}

In general the quadratic base equations are not uniquely determined. They can be modified by multiples of the linear lifting equations, if such are present. The other source of non-uniqueness is the possibility of coordinate transformations using the pure rolling factors variables.

Theorem 3.4. Let P = ∑ p_{I,k}s^{⟨e,I⟩−b−k}t^{k}z^{I} define a divisor of type 2H −bR. It leads to quadratic base equations π_{1}, …, π_{b−1}. The coefficient p_{I,k} gives the following term in π_{m}. We write z^{I} = xy and assume that e_{x} ≥ e_{y}. I. If e_{x} < b then for m ≤ k the term is − ∑_{l=m}^{k} η_{k−l+m}ξ_{l}, while for m > k it

is

∑_{l=max(k+m−e_{y}+1,k+1)}^{min(e_{x}−1,m−1)} η_{k−l+m}ξ_{l} .

II. If e_{x} ≥ b then for m ≤ k + b − e_{x} the term is − ∑_{l=m+e_{x}−b}^{k} η_{k−l+m}ξ_{l}, while for m > k + b − e_{x} it is

∑_{l=max(k+m−e_{y}+1,k+1)}^{min(e_{x}−b+m−1,k+m−1)} η_{k−l+m}ξ_{l} .

Furthermore, if e_{x} ≥ b the e_{x} − b + 1 pure rolling factors deformations involving x contribute ρ_{0}ξ_{m} + ⋯ + ρ_{e_{x}−b}ξ_{m+e_{x}−b} to π_{m}.

Proof . We need explicit equations P_{m}. The monomial s^{e−x+e_{y}−b−k}t^{k}xy gives a rolling monomial x_{i(m)}y_{j(m)}, where i(m) + j(m) = k + m. Let i(0) = i, i(b) = i′, j(0) = j and j(b) = j′. We have to compute P′_{m}. Equation (S) gives

s^{b}P′_{b} − t^{b}P′_{0} = ∑_{l=1}^{j′−j} s^{e_{x}−k+j+l}t^{k+b−j−l}xη_{j+l} + ∑_{n=1}^{i′−i} s^{e_{y}−k+i+n}t^{k+b−i−n}yξ_{i+n} ,

which we rewrite as

s^{b}P′_{b} − t^{b}P′_{0} = ∑_{l=j+1}^{j′} s^{e_{x}−k+l}t^{k+b−l}xη_{l} + ∑_{l=i+1}^{i′} s^{e_{y}−k+l}t^{k+b−l}yξ_{l} .

Case I: e_{x} < b. The condition k + b ≤ e_{x} + e_{y} implies k < e_{y}. We solve for P′_{0}:

P′_{0} = − ∑_{l=j+1}^{k} x_{k−l}η_{l} − ∑_{l=i+1}^{k} y_{k−l}ξ_{l} .

For the P′_{m} we formally write the formula

P′_{m} = − ∑_{l=j+1}^{k} x_{k−l+m}η_{l} − ∑_{l=i+1}^{k} y_{k−l+m}ξ_{l} + ∑_{l=j+1}^{j(m)} x_{k−l+m}η_{l} + ∑_{l=i+1}^{i(m)} y_{k−l+m}ξ_{l} .

This expression can involve non-existing x or y variables: for y this happens if k − l + m > e_{y}, or l < m + k − e_{y}. The terms in the two sums involving y cancel. If i(m) < k, then the smallest non-canceling term has l = i(m) + 1 and i(m) + 1 ≥ i(m) + j(m) − e_{y} = k + m − e_{y}. If i(m) > k we a sum of positive terms starting with k + 1. If k < l < m + k − e_{y} then our monomial contributes to the lifting conditions, and we can leave out this term. The sum therefore now starts at max(k + 1, m + k − e_{y}). Keeping this in mind we determine the term in the base equation π_{m} from the formal formula. To this end we change the summation variable in the sums containing x-variables and arrive, using i(m) + j(m) = k + m, at

− ∑_{l=m}^{m+i−1} ξ_{l}η_{k−l+m} − ∑_{l=i+1}^{k} η_{k−l+m}ξ_{l} + ∑_{l=i(m)}^{m+i−1} ξ_{l}η_{k−l+m} + ∑_{l=i+1}^{i(m)} η_{k−l+m}ξ_{l} −ξ_{i(m)}η_{j(m)}

= − ∑_{l=m}^{m+i−1} ξ_{l}η_{k−l+m} − ∑_{l=i+1}^{k} η_{k−l+m}ξ_{l} + ∑_{l=i+1}^{m+i−1} η_{k−l+m}ξ_{l} .

If k ≥ m the terms from l = m to l = k occur twice with a minus sign and once with a plus. Otherwise all negative terms cancel, but we have to take the lifting conditions into account.

Case II: e_{x} ≥ b. Now there are e_{x} − b + 1 pure rolling factors deformations present: we can perturb P_{m} with ρ_{0}x_{m} +⋯+ρ_{e_{x}−b}x_{m+e_{x}−b}. These contribute ρ_{0}ξ_{m} + ⋯ + ρ_{e_{x}−b}ξ_{m+e_{x}−b} to the equation π_{m}. We can roll using only the x variable: x_{i+m}y_{j}, with i + j = k and i + b ≤ e_{x}. We take i = k if k + b ≤ e_{x} and i = e_{x} − b otherwise. We get

s^{b}P′_{b} − t^{b}P′_{0} = ∑_{l=i+1}^{i+b} s^{e_{y}−k+l}t^{k+b−l}yξ_{l} .

We solve:

P′_{0} = − ∑_{l=i+1}^{k} py_{k−l}ξ_{l}

and

P′_{m} = − ∑_{l=i+1}^{k} y_{k−l+m}ξ_{l} + ∑_{l=i+1}^{i+m} y_{k−l+m}ξ_{l} .

Again if k < l < m + k − e_{y} our monomial contributes to the lifting conditions, and the sum starts at max(k + 1, m + k − e_{y}). We get as contribution to π_{m}

− ∑_{l=i+1}^{k} η_{k−l+m}ξ_{l} + ∑_{l=i+1}^{i+m−1} η_{k−l+m}ξ_{l} .

Taking the lifting conditions and our choice of i into account we get the statement of the theorem. □

Example 3.5: Case I. Let b = 7, e_{x} = 5 and e_{y} = 4. Consider the equation P = (p_{0}s^{2} + p_{1}st + p_{2}t^{2})xy. This leads to the following six equations:

π_{1} =  − p_{2}(ξ_{1}η_{2} + ξ_{2}η_{1})
π_{2} = p_{0}ξ_{1}η_{1}  − p_{2}ξ_{2}η_{2}
π_{3} = p_{0}(ξ_{1}η_{2} + ξ_{2}η_{1})  + p_{1}ξ_{2}η_{2}
π_{4} = p_{0}(ξ_{1}η_{3} + ξ_{2}η_{2}) + ξ_{3}η_{1} + p_{1}(ξ_{2}η_{3} + ξ_{3}η_{2}) + p_{2}ξ_{3}η_{3}
π_{5} = p_{0}(ξ_{2}η_{3} + ξ_{3}η_{2}) + ξ_{4}η_{1} + p_{1}(ξ_{3}η_{3} + ξ_{4}η_{2}) + p_{2}ξ_{4}η_{3}
π_{6} = p_{0}(ξ_{3}η_{3} + ξ_{4}η_{2})  + p_{1}ξ_{4}η_{3}

If we write a matrix with the coefficients of the p_{i} in the columns with rows coming from the equations π_{m} we find that the first k + 1 rows form a skew symmetric matrix. This is due to the specific choices made in the above proof. One can also get any other block to be skew symmetric by using the lifting conditions. In this example they are p_{0}η_{1}+p_{1}η_{2}+p_{2}η_{3} = 0, p_{0}ξ_{1}+p_{1}ξ_{2}+p_{2}ξ_{3} = 0 and p_{0}ξ_{2} + p_{1}ξ_{3} + p_{2}ξ_{4} = 0. From the skew symmetry we can conclude:

Proposition 3.6. If e_{y} ≤ e_{x} < b then the b − 1 equations π_{m} coming from the equation P = (∑_{j=0}^{k} p_{j}s^{k−j}t^{k})xy, where b + k = e_{x} + e_{y}, satisfy b − k − 1 linear relations ∑_{j=0}^{k} p_{j}π_{i+j} = 0, for 0 < i < b − k.

Example 3.7: case II. Let b = 4, e_{x} = 5 and e_{y} = 3. Consider the equation P = (p_{0}s^{4} + p_{1}s^{3}t + p_{2}s^{2}t^{2} + p_{3}st^{3} + p_{4}t^{4})xy. This leads to the following three equations:

\begin{matrix}π_{1} = ρ_{0}ξ_{1} + ρ_{1}ξ_{2} & − p_{2}ξ_{2}η_{1} − p_{3}(ξ_{2}η_{2} + ξ_{3}η_{1})p_{4}(ξ_{3}η_{2} + ξ_{4}η_{1}) \\ π_{2} = ρ_{0}ξ_{2} + ρ_{1}ξ_{3} & + p_{0}ξ_{1}η_{1} + p_{1}ξ_{2}η_{1} − p_{3}ξ_{3}η_{2} − p_{4}ξ_{4}η_{2} \\ π_{3} = ρ_{0}ξ_{3} + ρ_{1}ξ_{4} & +p_{0}(ξ_{1}η_{2} + ξ_{2}η_{1}) + p_{1}(ξ_{2}η_{2} + ξ_{3}η_{1}) + p_{2}ξ_{3}η_{2} \end{matrix}

(3.8) Hyperelliptic cones (cf. [Stevens 1996]). Let X be the cone over a hyperelliptic curve C embedded with a line bundle L of degree d ≥ 2g + 3. Then dim T_{X}^{1}(−1) = 2g + 2. The curve lies on a scroll of degree d − g − 1 as curve of type 2H − (d − 2g − 2)R. The number of rolling factors equations is d − 2g − 3, so we have at least as many equations as variables if d > 4g + 4. In that case only conical deformations exist, so all deformations in negative degree are obstructed. The easiest case to describe is L = ng_{2}^{1}. The curve C has an affine equation y^{2} = ∑_{k=0}^{2g+2} p_{k}t^{k}, which gives the bihomogeneous equation (∑_{k=0}^{2g+2} p_{k}s^{2g+2−k}t^{k})x^{2}− y^{2} = 0. The line bundle L embeds C in a scroll S(n, n − g − 1), and there are 2n−2g −1 rolling factors equations P_{m}, coming from p(s, t)x^{2} −y^{2}. The lifting matrix is a block diagonal matrix with the y-block equal to −2I_{n−g−2}, and the x-block a (n − 2g − 3) × (n − 1) matrix, so the dimension of the space of lifting deformations of the scroll is 2g + 2 if n ≥ 2g + 3. If n ≤ 2g + 3, the x-block is not present, and all n − 1 ξ-deformations lift. Furthermore there are 2g + 3 − n pure rolling factors deformations. This shows again that dim T_{X}^{1}(−1) = 2g + 2.

Proposition 3.9. If n ≥ 2g + 3 the base space in negative degrees is a zerodimensional complete intersection of 2g + 2 quadratic equations.

Proof . We may assume that the highest coefficient p_{2g+2} in p(s, t) equals 1. The lifting equations allow now to eliminate the variables ξ_{2g+3}, …, ξ_{n−1}. The base equations π_{m} involve only the ξ_{i} and are therefore not linearly independent. Because p_{2g+2} = 1 we can discard all π_{m} with m > 2g + 2. The first 2g +2 equations involve only the first 2g +2 variables. This shows that we have the same system of equations for all n ≥ 2g + 3. As we know that there are no deformations over a positive dimensional base, we conclude that the base space is a complete intersection of 2g + 2 equations. □

Remark 3.10. The fact that the system of equations above defines a complete intersection can also be seen directly. In fact we have the following result:

Lemma 3.11. The system of e = b−1 equations π_{m} in e−1 variables ξ_{i} coming from one polynomial P_{b−2}(s, t)x^{2} is a zero-dimensional complete intersection if and only if P_{b−2}(s, t) has no multiple roots.

Proof . First we note that there are only b − 2 linearly independent equations. We put ξ_{i} = s^{b−i−1}t^{i}. Then

π_{m} = ∑_{k=0}^{b−2}(m − k − 1)p_{k}s^{2b−k−m−2}t^{k+m−2}

= s^{b−m}t^{m−2}(∑(b − 2 − k)p_{k}s^{b−2−k}t^{k} + (m + 1 − b) ∑ p_{k}s^{b−2−k}t^{k}) .

The form P (s, t) has multiple roots if and only if P (s, t) and s\frac{∂}{∂s}P (s, t) have a common zero (s_{0} :t_{0}). Then ξ_{i} = s_{0}^{b−i−1}t_{0}^{i} is a nontrivial solution to the system of equations. We show the converse by induction. One first checks that a linear transformation in (s: t) does not change the isomorphism type of the ideal. We apply a transformation such that s = 0 is a single root of P, so p_{0} = 0 but p_{1} ≠ 0. The equations π_{2}, …, π_{b−1} now do not involve the variable ξ_{1} and are by the induction hypotheses a complete intersection in e − 2 variables, so their zero set is the ξ_{1}-axis with multiple structure. The equation π_{1} has the form −p_{1}ξ_{1}^{2} + …, so the whole system has a zero-dimensional solution set. □

Remark 3.12. For deg L = 4g + 4 the base space is a cone over 2^{2g+1} points in a very special position: there exist 2g + 2 hyperplanes {l_{i} = 0} such that the base is given by l_{i}^{2} = l_{j}^{2} [Stevens 1996]. We can make this more explicit in the case L = (2g + 2)g_{2}^{1}. Again the y-block of the lifting matrix is a multiple of the identity, but now there is also one rolling factors deformation parameter ρ. More generally, we look the equations coming from p(s, t)x^{2} with deg p = b = e. We get base equations Π_{m} = ρξ_{m} + π_{m}, where π_{m} is a quadratic equation in the ξ-variables only. One solution is clearly ξ_{i} = 0 for all i. To find the others we eliminate ρ:

Rank ( \begin{matrix}π_{1} & π_{2} & … & π_{e−1} \\ ξ_{1} & ξ_{2} & … & ξ_{e−1} \end{matrix} ) ≤ 1 .  (∗∗)

The equations Π_{m} can be changed by changing ρ, but this system is independent of such changes. Write inhomogeneously p(t) = p_{0} +p_{1}t+…+p_{e−1}t^{e−1} + t^{e} = ∏(t − α_{i}), where the α_{i} are the roots of p(t).

Lemma 3.13. The e points P_{i} = (1 : α_{i} : α_{i}^{2} : ⋯: α_{i}^{e−2}) are solutions to the system (∗∗).

Proof . Let α be a root of p and insert ξ_{i} = α^{i−1} in the system (∗∗). We simplify the matrix by column operations: subtract α times the jth column from the (j + 1)st column, starting at the end. The matrix has clearly rank 1, if π_{j+1}(α) − απ_{j}(α) = 0, where π_{j}(α) is the result of substituting ξ_{i} = α^{i−1} in the equation π_{j}. The coefficient p_{k} occurs in π_{j}(α) in the term lp_{k}α^{j+k−2} for some integer l, and in the term (l + 1)p_{k}α^{j+k−1} in π_{j+1}(α). Therefore π_{j+1}(α) − απ_{j}(α) = − ∑ p_{k}α^{j+k−1} = −α^{j−1}p(α) = 0. □

The remaining solutions are found in the following way. Divide the set of roots into two subsets I and J. The points P_{i} lie on a rational normal curve. Therefore the points P_{i} with i ∈ I span a linear subspace L_{I} of dimension |I| − 1.

Claim. The intersection point P_{I} := L_{I} ∩ L_{J} is a solution to (∗∗).

The proof is a similar but more complicated computation. We determine here only the point P_{I}. The condition that the point ∑_{i∈I} λ_{i}P_{i} lies in L_{J} is that

Rank ( \begin{matrix}∑ λ_{i} & … & ∑ λ_{i}α_{i}^{e−2} \\ 1 & … & α_{j_{1}}^{e−2} \\ ⋮ & & ⋮ \\ 1 & … & α_{j_{|J|}}^{e−2} \end{matrix} ) = |J| .

We find the resulting linear equations on the λ_{i} by extending the matrix to a square matrix by adding |I| − 2 rows of points on the rational normal curve, for which we take roots. Then only two λ_{i} survive, and they come with a Vandermonde determinant as coefficient. Upon dividing by common factors we get (∏_{i≠i_{1},i_{2}}(α_{i_{1}} − α_{i}))λ_{i_{1}} + (∏_{i≠i_{1},i_{2}}(α_{i_{2}} − α_{i}))λ_{i_{2}} = 0. We multiply with α_{i_{1}} − α_{i_{2}}. Noting that ∏_{i≠i_{1}}(α_{i_{1}} − α_{i}) = p′(α_{i_{1}}) (with p′(t) the derivative of p(t)) we get p′(α_{i_{1}})λ_{i_{1}} = p′(α_{i_{2}})λ_{i_{2}}. We write out the equations for e = 5:

ρξ_{1} − p_{1}ξ_{1}^{2} − 2p_{2}ξ_{1}ξ_{2} − p_{3}ξ_{2}^{2} − 2p_{4}ξ_{2}ξ_{3} − p_{5}(2ξ_{2}ξ_{4} + ξ_{3}^{2})
ρξ_{2} + p_{0}ξ_{1}^{2} − p_{2}ξ_{2}^{2} − p_{4}ξ_{3}^{2} − 2p_{5}ξ_{3}ξ_{4}
ρξ_{3} + 2p_{0}ξ_{1}ξ_{2} + p_{1}ξ_{2}^{2} − p_{3}ξ_{3}^{2} − p_{5}ξ_{4}^{2}
ρξ_{4} + p_{0}(2ξ_{1}ξ_{3} + ξ_{2}^{2}) + 2p_{1}ξ_{2}ξ_{3} + p_{2}ξ_{3}^{2} + 2p_{3}ξ_{3}ξ_{4} + p_{4}ξ_{4}^{2}

Let α be a root of p_{0} + p_{1}t + p_{2}t^{2} + p_{3}t^{3} + p_{4}t^{4} + t^{5}, and β, …, ε the remaining roots. Write σ′_{i} for the ith symmetric function of these four roots. Then a solution is ξ_{i} = α^{i−1}, ρ = α^{4} − α^{3}σ′_{1} − α^{2}σ′_{2} − ασ′_{3} + σ′_{4}. Given two roots α and β we get a solution ξ_{i} = (γ − β)(δ − β)(ε − β)α^{i} + (α − γ)(α − δ)(α − ε)β^{i}. To write ρ we set μ = (γ − β)(δ − β)(ε − β), λ = (α − γ)(α − δ)(α − ε) and σ′′_{i} the ith symmetric function in γ, δ and ε. Then and ρ = μ(α^{4} − α^{2}(α + 2β)σ′′_{1} − α^{2}σ′′_{2} − ασ′′_{3}) + λ(β^{4} − β^{2}(α + 2α)σ′′_{1} − β^{2}σ′′_{2} − βσ′′_{3}). The hyperplane through (1 : 0 : 0 : 0 : 0), P_{γ}, P_{δ} and P_{ε} is l_{αβ}^{−} = σ′′_{3}ξ_{1} − σ′′_{2}ξ_{2} + σ′′_{1}ξ_{3} − ξ_{4}. In it lie also P_{γδ}, P_{γε}, P_{δε} and P_{αβ}. The hyperplane containing the remaining points is l_{αβ}^{+} = ρ−(α+β)l_{αβ}^{−} +2σ′′_{3}ξ_{2} +2αβξ_{3}. We put l_{a} = ρ−2σ′_{4}ξ_{1} +2σ′_{3}ξ_{2} +2ασ′_{1}ξ_{3} −2αξ_{4}. Then l_{α}^{2} − l_{β}^{2} = 4(α − β)l_{αβ}^{−}l_{αβ}^{+}.

4. Tetragonal curves. An extension of a canonical curve yields a surface with the given canonical curve as hyperplane section. Surfaces with canonical hyperplane sections were studied in Dick Epema’s thesis [Epema 1983]. Only a limited list of surfaces can occur.

Theorem 4.1([Epema 1983], Cor. I.5.5 and Cor. II.3.3). Let W be a surface with canonical hyperplane sections. Then one of the following holds:

(a) W is a K3 surface with at most rational double points as singularities,

(b) W is a rational surface with one minimally elliptic singularity and possibly rational double points,

(c) W is a birationally ruled surface over an elliptic curve Γ with as nonrational singularities either

i) two simple elliptic singularities with exceptional divisor isomorphic to Γ, or

ii) one Gorenstein singularity with p_{g} = 2,

(d) W is a birationally ruled surface over a curve Γ of genus q ≥ 2 with one non-rational singularity with p_{g} = q+1, whose exceptional divisor contains exactly one non-rational curve isomorphic to Γ.

Case (c) occurs for bi-elliptic curves (see below). If we exclude them and curves of low genus on Del Pezzo surfaces, then all extensions of tetragonal curves are of rolling factors type. The surface W has therefore to occur in our classification of complete intersection surfaces on scrolls. In particular, K3 surfaces can only occur if b_{1} ≤ b_{2} + 4. This has consequences for deformations of tetragonal cones.

Proposition 4.2. Pure rolling factors deformations are always unobstructed. If e_{3} > 0 and b_{1} > b_{2} + 4 the remaining deformations are obstructed.

Proof . The first statement follows directly from the form of the equations. For the second we note that the total space of a nontrivial one-parameter deformation of a scroll with e_{3} > 0 is a scroll with e_{4} > 0. □

By taking hyperplane sections of a general element in each of the families of the classification we obtain for all g tetragonal curves with b_{1} ≤ b_{2} + 4 lying on K3 surfaces (with at most rational double points). To realize the other types of surfaces we give a construction, which goes back to [Du Val 1933]. His construction was generalized to the non-rational case in [Epema 1983]. In our situation we want a given curve to be a hyperplane section. A general construction for given hyperplane sections of regular surfaces is given in [Wahl 1998].

Construction 4.3. Let Y be a surface containing the curve C and let D ∈ | − K_{Y} | be an anticanonical divisor. Let Ỹ be the blow up of Y in the scheme Z = C ∩D. If the linear subsystem C′ of |C| with base scheme Z has dimension g, it associated map contracts D and blows down Ỹ to a surface \overline{Y} with C as canonical hyperplane section.

Let I_{Z} be the ideal sheaf of Z. Then we have the exact sequence

0 ⟶ O_{Y} ⟶ I_{Z}O_{Y} (C) ⟶ O_{C}(C − Z) ⟶ 0

and by the adjunction formula O_{C}(C − Z) = K_{C}. If h^{0}(I_{Z}O_{Y} (C)) = g + 1 then the map H^{0}(I_{Z}O_{Y} (C)) ⟶ H^{0}(K_{C}) is surjective, a condition which is automatically satisfied if Y is a regular surface. This yields that the special hyperplane section is the curve C in its canonical embedding. Suppose that Y is not regular. By Epema’s classification Y is then a birationally ruled surface, over a curve Γ of genus q. Let C̃ be the strict transform of C on Ỹ and \overline{C} its image on \overline{Y}. Then H^{0}(I_{Z}O_{Y} (C)) = H^{0}(O_{Y}(C̃)) = H^{0}(O\overline{Y} (\overline{C})). We look at the exact sequence

0 ⟶ O\overline{Y} ⟶ O\overline{Y} (\overline{C}) ⟶ O\overline{C}(\overline{C}) = K_{C} ⟶ 0 .

We compute H^{1}(O\overline{Y}) with the spectral sequence for the map π: Ỹ ⟶ \overline{Y}. This gives us the long exact sequence

0 ⟶ H^{1}(O\overline{Y}) ⟶ H^{1}(O_{Y}) ⟶ H^{0}(R^{1}π_{∗}O_{Y}) ⟶ H^{2}(O\overline{Y}) ⟶ 0

in which dim H^{1}(O_{Y}) = q. We choose D in such a way that the composed map H^{1}(O_{Γ}) ⟶ R^{1}π_{∗}O_{Y} ⟶ H^{1}(O_{D}), where D̃ is the exceptional divisor of the map π, is injective. Then the map H^{0}(I_{Z}O_{Y} (C)) ⟶ H^{0}(K_{C}) is surjective. To apply the construction we need a surface on which the curve C lies. In the tetragonal case a natural candidate is the surface Y of type 2H − b_{1}R on the scroll. We first assume that e_{1} < b_{1}, so there are no pure rolling factors deformations coming from the first equation on the scroll. The canonical divisor of the scroll S is −3H + (b_{1} + b_{2})R [Schreyer 1986, 1.7]. So an anticanonical divisor on Y is of type H − b_{2}R. Let T = τ_{e_{1}−b_{2}}(s, t)x + τ_{e_{2}−b_{2}}(s, t)y + τ_{e_{3}−b_{2}}(s, t)z be the equation of such a divisor. Sections of I_{Z}O_{Y} (C) are Q (which defines C), and x_{i}T = s^{e_{1}−i}t^{i}xT, y_{i}T and z_{i}T. With coordinates (t: x_{i} : y_{i} : z_{i}) on ℙ^{g} we get by rolling factors b_{2} + 1 equations Q̃_{m} from the relation Q(τ_{e_{1}−b_{2}}(s, t)x + τ_{e_{2}−b_{2}}(s, t)y+τ_{e_{3}−b_{2}}(s, t)z) = (Q_{1,1}x^{2}+⋯+Q_{3,3}z^{2})T. As t is also a coordinate on the four-dimensional scroll, which is the cone over S, we can write the equation on the scroll as

Q_{1,1}x^{2} + ⋯ + Q_{3,3}z^{2} − (τ_{e_{1}−b_{2}}x + ⋯ + τ_{e_{3}−b_{2}}z)t .

We analyze the resulting singularities. If Y is a rational surface, we have a anticanonical divisor D which has arithmetic genus 1, giving a minimally elliptic singularity on the total space of the deformation. If Y is a ruled surface over a hyperelliptic curve Γ, then D passes through the double locus. This gives an exceptional divisor with Γ as only non-rational curve.

Example 4.4. Let (e_{1}, e_{2}, e_{3}; b_{1}, b_{2}) = (3n−2, 2n−2, n−2; 4n−4, 2n−4). If the coefficient of xz does not vanish, we may bring the equation P onto the form xz − y^{2}. The second equation has the form z^{2} + q_{n}zy + q_{2n}zx + q_{3n}xy + q_{4n}x^{2} from which z may be eliminated to obtain a quartic equation for y. The case of a cyclic curve y^{4} + q_{4n}x^{4} is a special instance. The equation P gives a square lifting matrix in which the antidiagonal blocks are square unit matrices. Therefore the only deformations are pure rolling factors deformations, coming from the second equation, in number (n + 3) + 3 = \frac{g+3}{6} + 6. We have T = τ_{n+2}x + τ_{2}y. The section (0 : 0 : 1) is always a component of D. If t_{2} ≢ 0 we have a cusp singularity, but if τ_{2} ≡ 0 the section occurs with multiplicity 2 in D. If however the coefficient of xz vanishes, the surface Y is singular. After a coordinate transformation its equation is y^{2} + p_{2n}x^{2}, the other equation being z^{2} + q_{3n}xy + q_{4n}x^{2}. In this case the lifting matrix has (up to a factor \frac{1}{2}) the following block structure

( \begin{matrix}Π & 0 & 0 \\ 0 & I & 0 \\ 0 & 0 & 0 \\ 0 & 0 & I \end{matrix} )

so there are 2n ξ-deformations, on which we have 4n−5 base equations coming from the equation P. Of these are only 2n linearly independent, defining a zero-dimensional complete intersection (see Lemma 3.11). These deformations are therefore obstructed, leaving us again with only the pure rolling factors deformations. The curve D consists of the double locus and in general 2n + 4 lines.

The same computation as above works for bielliptic cones. In that case one has a deformation of weight −2. The total space is a surface in weighted projective space ℙ(1, …, 1, 2). Replacing the deformation parameter t by t^{2} we get a surface in ordinary ℙ^{g}. This is a surface with two simple elliptic singularities. The most general surface of this type is the intersection of our elliptic cone with one dimensional vertex with the hypersurface given by

Q̃ = z^{2} + Q(x_{i}, y_{i}) + tl(x_{i}, y_{i}) + at^{2} ,

where l(x_{i}, y_{i}) is a linear form in the coordinates x_{i}, y_{i}. If the coefficient a vanishes, we get a surface with one singularity with p_{g} = 2. The construction above gives an equation of the form Q̃ = z^{2} +⋯+azt, which after a coordinate transformation becomes z^{2} + ⋯ − \frac{1}{4}a^{2}t^{2}.

Proposition 4.5. For bielliptic cones of genus g > 10 the only deformations of negative weight are pure rolling factors deformations.

Proof . Each infinitesimal deformation of the bielliptic cone induces an infinitesimal deformation of the cone over the projective cone over the elliptic curve. The same holds therefore for complete deformations of negative weight. It is well-known that the cone over an elliptic curve of degree at least 10 has only obstructed deformations of negative weight. Therefore the deformation of the elliptic cone is trivial and the only possibility is to deform the last quadratic equation. □

On the other hand, non-scrollar extension do occur for bielliptic curves with g ≤ 10 and for tetragonal curves on Del Pezzo surfaces.

Example 4.6. A bielliptic curve of genus 10 lies on the projective cone over an elliptic curve of degree 9. Such a cone is can be smoothed to the triple Veronese embedding of ℙ^{2}. Let W be a K3 surface of degree 2, a double cover of ℙ^{2} branched along a sextic curve. We re-embed W with |3L|, where L is the pull-back of a line on ℙ^{2}. The image lies on the cone over the Verones embedding. A hyperplane section through the vertex of the cone is a bielliptic curve, whereas the general hyperplane section has a g_{6}^{2}. This example, due to [Donagi–Morrison 1989], is the only case where the gonality of smooth curves in a base-point-free ample linear system on a K3 surface is not constant [Ciliberto– Pareschi 1995].

Now we look at the case that also the first set of equations admit pure rolling factors deformations.

Lemma 4.7. If e_{1} ≥ b_{1} then e_{1} ≤ b_{1} + 2 and b_{1} ≤ b_{2} + 4.

Proof . Under the assumption e_{1} ≥ b_{1} we have e_{2} + e_{3} − 2 ≤ b_{2} ≤ 2e_{3} so e_{2} ≤ e_{3} + 2 and b_{1} ≤ 2e_{2} ≤ e_{2} + e_{3} + 2 ≤ b_{2} + 4. Furthermore e − 1 = b_{1} + b_{2} + 2 − e_{2} − e_{3} ≤ b_{1} + 2. □

It is now easy to list all 18 possibilities, ranging from (2e + 2, e, e; 2e, 2e) to (2e + 4, e + 2, e; 2e + 4, 2e). A look at the table of tetragonal K3 surfaces reveals that all possibilities are realizable as special sections of K3-surfaces; e.g., the hyperplane section x_{e+2} = y_{0} of a K3 with invariants (e + 2, e + 2, e + 2, e; 2e + 4, 2e) yields the last case. On the other hand, every family of K3 surfaces contains degenerate elements with singularities of higher genus. Those can be constructed with Epema’s construction and in fact he gives rather complete results for quartic hypersurfaces [Epema 1983]. The classification of such surfaces is due to [Rohn 1884] and is quite involved. In those cases the rational or ruled surfaces on which the canonical curve lies are not evident. For pure rolling factors extensions the situation is better; in fact, we can make the following simple observation.

Proposition 4.8. Let W be a pure rolling factors extension of tetragonal curve, which is not bi-elliptic. It lies on the cone over the 3-dimensional scroll S with vertex in p = (0 : …: 0 : 1) and the projection from the point p yields a surface Y ⊂ ℙ^{g−1} on which C lies.

Example 4.9. If b_{1} > e_{1} then X lies on the cone over the surface Y on the scroll and the projection is just this surface Y, so we get the construction described above.

Example 4.10. Consider the curve with invariants (8, 4, 2; 8, 4). In general a pure rolling factors extension leads to a K3-surface (with an ordinary double point). It is the case e = 3 of (e + 5, e + 1, e − 1, e − 3; 2e + 2, 2e − 1) from the table; the singularity appears because the section (0 : 0 : 0 : 1) is contracted. To find the equation of Y on the scroll we have to eliminate the last coordinate w. The deformed equation P̃ is P + axw, while Q̃ = Q + byw + c_{2}(s, t)xw with a and b nonzero constants. The equation of Y is therefore (by+c_{2}(s, t)x)P −axQ, which defines a divisor of type 3H − (b_{1} + b_{2})R on the scroll.

Example 4.11: the case (2e + 2, e, e; 2e, 2e). We first derive a normal form for the equations P and Q. We start with the restriction to x = 0. We have a pencil of quadrics so we may choose the first equation as y^{2} and the second as z^{2}. We get:

\begin{matrix}P: & y^{2} + p_{e+2}xz + p_{2e+4}x^{2} \\ Q: & z^{2} + q_{e+2}xy + q_{2e+4}x^{2} . \end{matrix}

There are 3 + 3 pure rolling factors deformations:

\begin{matrix}P̃: & P + (ρ_{0}s^{2} + ρ_{1}st + ρ_{2}t^{2})x \\ Q̃: & Q + (τ_{0}s^{2} + τ_{1}st + τ_{2}t^{2})x . \end{matrix}

If the polynomials ρ := ρ_{0}s^{2} + ρ_{1}st + ρ_{2}t^{2} and τ := τ_{0}s^{2} + τ_{1}st + τ_{2}t^{2} are proportional, so λρ + μτ = 0, then the surface Y is the surface λP + μQ = 0 from the pencil. In general the anticanonical divisor D contains the two sections given by x = 0, λy^{2} + μz^{2} = 0 and the singularity on the deformation is a cusp singularity. If ρ and τ have 0 ≤ γ < 2 roots in common, the projected surface is a divisor of type 2H − (2e − 2 + γ)R. In general we get a simple elliptic singularity. To describe the remaining deformations we look at the lifting matrix, which is a block matrix

( \begin{matrix}0 & 2I & 0 \\ Π_{e+2} & 0 & 0 \\ Ξ_{e+2} & 0 & 0 \\ 0 & 0 & 2I \end{matrix} )

of size (4e − 4) × (4e − 1). Its rank is 2e − 2 if p_{e+2} and q_{e+2} both vanish identically, and lies between 3e − 3 and 4e − 4 otherwise. The solution space has dimension γ ≥ 3 with strict inequality iff the polynomials p_{e+2} and q_{e+2} have γ roots in common. The η and ζ deformations vanish. Therefore the base equations depend only on p_{2e+4} and q_{2e+4}. They are 2(2e − 1) quadratic equations on 2e + 1 + 6 variables, which may or may not have solutions.

We now turn to the other deformations in general. A dimension count shows that the general tetragonal curve of genus g > 15 cannot lie on a K3 surface, so the deformations are obstructed. For a general tetragonal cone we have that dim T_{X}^{1} = 9. There are (b_{1} − 1) + (b_{2} − 1) = g − 7 quadratic base equations. Compare this with the dimension of T^{2}:

Theorem 4.12([Wahl, Thm. 5.9]). Let X be a tetragonal cone with e_{3} > 0. Then dim T_{X}^{2}(−k) = 0 for k > 2 and dim T_{X}^{2}(−k) = g − 7 if b_{2} > 0. If b_{2} = 0, then dim T_{X}^{2}(−k) = 2(g − 6).

In particular, if g > 15 we have more equations than variables and in general there are no solutions. For special moduli solutions do exist and one expects in general exactly one solution.

(4.13) The case g = 15. Consider the most general situation, of equal invariants: e_{1} = e_{2} = e_{3} = 4, b_{1} = b_{2} = 5. In this case there are no pure rolling factor deformations and no lifting conditions.

Proposition 4.14. The general tetragonal curve with e_{1} = e_{2} = e_{3} = 4, b_{1} = b_{2} = 5 is hyperplane section of 256 different K3 surfaces.

Proof . We have 8 homogeneous quadratic equations in 9 variables, which define a complete intersection of degree 2^{8}. We give an explicit example. Take the curve, given by the equations

(s^{3} + t^{3})x^{2} + (s^{3} + 2t^{3})y^{2} + (s^{3} − 2t^{3})z^{2}
(s^{2} + t^{2})(s − t)x^{2} + s^{2}(s + t)y^{2} + t^{3}z^{2}

on the scroll. The base equations are formed according to Thm. 3.4. One computes that indeed we have a complete intersection, which is non-singular. □

It is very difficult to find solutions to such equations, and I have not succeeded to do so in the specific example. Note that the absence of mixed terms in x, y and z on the scroll means that the automorphism group of the curve has order at least eight and it operates on the base space: given one solution one finds three other ones by multiplying all ξ_{i} or all ζ_{i} by −1.

Remark 4.15. Alternatively one can start with a K3 surface and take a general hyperplane section. Therefore we look at complete intersections of two surfaces of type 2H − 5R on a scroll of type (3, 3, 3, 3). Such a K3 surface can have infinitesimal deformations of negative weight (which are always obstructed). The lifting matrix for the K3 has size 8 × 8. The equations P and Q on the scroll are pencils of quadrics. In general such a pencil has 4 singular fibres and by taking a suitable linear combination we may suppose that P has the form

sX^{2} + tY^{2} + (s + t)Z^{2} + (s − t)W^{2} .

The polynomial Q is then a general pencil with 20 coefficients, of which one can be made to vanish by subtracting a multiple of P. This shows that these K3 surfaces depend on 18 moduli. Let Q = (a_{11}s + b_{11}t)X^{2} + 2(a_{12}s + b_{12}t)XY + … + (a_{44}s + b_{44}t)W^{2}. Then the lifting matrix is

2 ( \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & −1 \\ a_{11} & b_{11} & a_{12} & b_{12} & a_{13} & b_{13} & a_{14} & b_{14} \\ a_{12} & b_{12} & a_{22} & b_{22} & a_{23} & b_{23} & a_{24} & b_{24} \\ a_{13} & b_{13} & a_{23} & b_{23} & a_{33} & b_{33} & a_{34} & b_{34} \\ a_{14} & b_{14} & a_{24} & b_{24} & a_{34} & b_{34} & a_{44} & b_{44} \end{matrix} ) .

For nonsingular K3 surfaces this matrix has at least rank 5, and it is possible to write down examples with exactly rank 5. Rank 4 can be realized with surfaces with isolated singularities. An explicit example (with a slightly different basis for the pencil) is

\begin{aligned}P & = sX^{2} + tY^{2} + (s + t)Z^{2} \\ Q & = sX^{2} − tY^{2} + (s − t)W^{2} \end{aligned}

with ordinary double points at sX = tY = (s + t)Z = (s − t)W = 0. The hyperplane section X_{3} + Z_{2} + W_{1} + Y_{0} = t^{3}X + s^{2}tZ + st^{2}W + s^{3}Y does not pass through the singular points and defines a smooth tetragonal curve with e_{i} = 4. The base space for this curve is still a complete intersection, but the line corresponding to the singular K3 surface is a multiple solution.

(4.16) The case g = 16. The curves lying on a K3 form a codimension one subspace in the moduli space of tetragonal curves of genus g = 16. In terms of the coefficients of the equations of the scroll one gets an equation of high degree. It makes no sense to write it. We will not study the most general case (5, 4, 4; 6, 5) but (5, 5, 3; 6, 5). These curves form a codimension two subspace in moduli. The computations will show that the condition of being a hyperplane section has again codimension one. The lifting matrix need not have full rank. We have b_{1} = 2e_{3}, and the g_{4}^{1} can be composed. Suppose that the coefficient of z^{2} in the first equation on the scroll does not vanish. With a coordinate transformation we may assume that the equation has the form z^{2} + P_{4}(s, t; x, y) with P_{4} of degree 4 in (s : t) and quadratic in (x: y). Then we can take Q to be without z^{2} term. Let q_{0;1}s^{3} + … + q_{3;1}t^{3} be the coefficient of xz and q_{0;2}s^{3} + … + q_{3;2}t^{3} that of yz. The rows of the 3 × 8 lifting matrix come only from the monomial z:

( \begin{matrix}0 & 0 & 0 & 0 & | & 0 & 0 & 0 & 0 & | & 1 & 0 \\ 0 & 0 & 0 & 0 & | & 0 & 0 & 0 & 0 & | & 0 & 1 \\ q_{0;1} & q_{1;1} & q_{2;1} & q_{3;1} & | & q_{0;2} & q_{1;2} & q_{2;2} & q_{3;2} & | & 0 & 0 \end{matrix} )

The matrix has rank 3 if some q_{i,j} does not vanish, but rank 2 if they all vanish; then the surface {Q = 0} has a singular line. The deformation variables ζ_{1}, ζ_{2} vanish. We have two pure rolling factors deformations ρ_{1} and ρ_{2} in the second set of additional equations, and there are scrollar deformations ξ_{1}, …, η_{4}. Between those exist a linear relation given by the third line of the matrix. The equations for the base can be written down independently of this linear relation, because the ζ_{i} vanish. We give a specific example: z^{2} + t^{4}y^{2} + (s^{4} + t^{4})x^{2} and (s^{5} + t^{5})x^{2} + (s^{5} − t^{5})y^{2} + q_{1}(s, t)xz + q_{2}(s, t)yz. We get the following nine equations:

−2ξ_{2}ξ_{3} − 2ξ_{1}ξ_{4} − 2η_{2}η_{3} − 2η_{1}η_{4}
ξ_{1}^{2} − ξ_{3}^{2} − 2ξ_{2}ξ_{4} − η_{3}^{2} − 2η_{2}η_{4}
2ξ_{1}ξ_{2} − 2ξ_{3}ξ_{4} − 2η_{3}η_{4}

2ξ_{1}ξ_{3} + ξ_{2}^{2} − ξ_{4}^{2} − η_{4}^{2}
2ξ_{1}ξ_{4} + 2ξ_{2}ξ_{3}

ρ_{1}ξ_{1} + ρ_{2}η_{1} − ξ_{3}^{2} − 2ξ_{2}ξ_{4} + η_{3}^{2} + 2η_{2}η_{4} ρ_{1}ξ_{2} + ρ_{2}η_{2} + ξ_{1}^{2} + η_{1}^{2} − 2ξ_{3}ξ_{4} + 2η_{3}η_{4} ρ_{1}ξ_{3} + ρ_{2}η_{3} + 2ξ_{1}ξ_{2} + 2η_{1}η_{2} − ξ_{4}^{2} + η_{4}^{2} ρ_{1}ξ_{4} + ρ_{2}η_{4} + 2ξ_{1}ξ_{3} + ξ_{2}^{2} + 2η_{1}η_{3} + η_{2}^{2}

Also in general we have 5 equations π_{m} and 4 equations ρ_{1}ξ_{m} + ρ_{2}η_{m} + χ_{m}. The pure rolling factors equations are never obstructed. We have as solution to the equations therefore the (ρ_{1}, ρ_{2})-plane with a non reduced structure. Given a general value of (ρ_{1}, ρ_{2}) we can eliminate say the η_{i} variables. We are then left with 5 equations π_{i} depending only on the x_{i}. Their quadratic parts satisfy a relation with constant coefficients, but even more is true: this relation can be lifted to the equations themselves. So the component has multiplicity 16. The general fibre over the reduced component has a simple elliptic singularity of degree 10. To find the other solutions we eliminate ρ_{1} and ρ_{2}. This gives the condition

Rank ( \begin{matrix}χ_{1} & χ_{2} & χ_{3} & χ_{4} \\ ξ_{1} & ξ_{2} & ξ_{3} & ξ_{4} \\ η_{1} & η_{2} & η_{3} & η_{4} \end{matrix} ) ≤ 2

which defines a codimension 2 variety of degree 11. In general the 5 equations π_{m} cut out a subset of codimension 7 and degree 352. But if

Rank ( \begin{matrix}ξ_{1} & ξ_{2} & ξ_{3} & ξ_{4} \\ η_{1} & η_{2} & η_{3} & η_{4} \end{matrix} ) ≤ 1  (R)

the full equations have only solutions in the (ρ_{1}, ρ_{2})-plane. Even if this rank condition defines a codimension 3 subspace, there are always solutions. To see this we set ξ_{i} = s^{4−i}t^{i−1}ξ and η_{i} = s^{4−i}t^{i−1}η. The equations π_{m} are satisfied if \frac{∂}{∂s}P_{4}(s, t; ξ, η) = 0 and \frac{∂}{∂t}P_{4}(s, t; ξ, η) = 0. This is the intersection of two curves of type (2, 3) on the scroll S_{3,3} ≅ ℙ^{1} × ℙ^{1} and there are 12 such intersection points. Those points give multiple solutions. One can compute that the rank of the Jacobi matrix of the system of equations (R) together with the π_{m} is five. By taking a suitable general example one finds that the multiplicity is in fact 4, and 48 is the degree of the solution of the system.

Proposition 4.17. The general tetragonal cone with invariants (5, 5, 3; 6, 5), which is composed with an involution of genus 4, has 302 smoothing components. The base space of a non-composed cone can be identified with a hyperplane section of the base of the corresponding composed one and only the smoothing components of lying in this hyperplane give smoothing components of the non-composed cone.

This means that for fixed polynomials P, Q the existence of smoothings depends on an equation of degree 302 in the eight variables q_{i;j}. For special values the number of smoothing components may go down. This happens in the specific example given, where the condition (R) gives two-dimensional ‘false’ solutions. Here there are only 238 smoothing components. Besides the hyperelliptic involution the curve has another automorphism which acts on the base space. The only solutions I have found are easy to see:

η_{1} = η_{2} = ξ_{3} = ξ_{4} = ξ_{1} + ξ_{2} = η_{3} + η_{4} = ρ_{1} + ξ_{1} = ρ_{2} + η_{4} = ξ_{1}^{2} − η_{4}^{2} = 0

We take ξ_{1} = η_{3} = ρ_{2} = δ and ξ_{2} = η_{4} = ρ_{1} = −δ. The total space is a surface on a scroll of type (4, 3, 3, 3) with bihomogeneous coordinates (W, X, Y, Z; s, t). We set Y_{i} = y_{i}, X_{i} = x_{i+2} and Z_{i} = z_{i} for i = 0, …, 3. The hyperplane section is δ = W_{2} + X_{0} + X_{1} + Y_{2} + Y_{3}, so if δ = 0 we have X = t^{2}x, Y = s^{2}y, Z = z and W = −(s+t)(x+y). The lifting equation is now q_{0;1} −q_{1;1} +q_{2;2} −q_{3;3} = 0. One computes that the surface is given by

2X^{2} + Y^{2} − 2(X − Y)W (s − t) + W^{2}(s − t)^{2} + Z^{2}

2Y^{2}s − XW (s^{2} − 2st + 2t^{2}) + Y W (2s^{2} − 2st + t^{2}) + W^{2}(s − t)(s^{2} − st + t^{2}) − XZ(sq_{2;2} + tq_{3;2}) − Y Z(sq_{0;1} + tq_{1;1}) − ZW (s^{2}q_{0;1} + st(q_{2;2} − q_{3;2}) + t^{2}q_{3;2})

This is a K3 surface with an A_{1}-singularity. For even more special values of the coefficients there may be higher dimensional smoothing components. This happens e.g. for P = z^{2}+t^{4}y^{2}+s^{4}x^{2} and the same Q as above, where the equations π_{m} have the solution ξ_{1} = ξ_{2} = η_{3} = η_{4} = 0, giving rise to an extra component of degree 15, which is the cone over three rational normal curves of degree 5. Then all tetragonal on Y have smoothings, but depending on the position of the hyperplane the number may increase.

(4.18) The case (b_{1}, b_{2}) = (8, 4). In this case there exist five families of K3surfaces, three of which have the maximal dimension 18. The general hyperplane section of the scroll S_{8,4,2,0} is a scroll S_{8,4,2} while for both S_{5,4,3,2} and S_{4,4,4,2} it is S_{5,5,4}. One computes that the tetragonal curves of type (2H − 8R, 2H − 4R) on S_{8,4,2} depend on 29 moduli and those on S_{5,5,4} depend on 34 moduli.

Proposition 4.19. The general tetragonal curve of type (8, 4, 2; 8, 4) has only pure rolling factors extensions. If the g_{4}^{1} is composed with an involution of genus 3, then there are in general 91 smoothing components not of this type.

Remark 4.20. The tetragonal curve can be a special hyperplane section of a K3 surface on S_{7,4,2,1}, S_{6,4,2,2}, S_{5,4,3,2} or S_{4,4,4,2}. Therefore the genericity assumption cannot be dropped.

Proof . After a coordinate we may assume that P has the form p_{8}x^{2}+y^{2}+p_{2}xz. The g_{4}^{1} is composed with an involution of genus 3 if and only if p_{2} ≡ 0. In that case Q may be taken in the form q_{12}x^{2}+q_{8}xy+z^{2}. That the curve is nonsingular implies that p_{8} has no multiple roots. If the g_{4}^{1} is not composed, the term z^{2} may be absent in Q, and p_{8} may have multiple roots. For the general curve this does not occur. We look therefore at curves given by

\begin{aligned}P & : p_{8}x^{2} + y^{2} + p_{2}xz \\ Q & : q_{12}x^{2} + q_{8}xy + z^{2} . \end{aligned}

The lifting matrix is a block matrix

( \begin{matrix}0 & 2I & 0 \\ Π & 0 & 0 \\ 0 & 0 & 2I \end{matrix} )

with Π giving the equations p_{2,0}ξ_{i} + p_{2,1}ξ_{i+1} + p_{2,2}ξ_{i+2} = 0. There is one pure rolling factors deformation for the first equation, and 5 + 1 for the second. The equation P leads to 7 base equations π_{m} in the 8 variables ρ, ξ_{1}, …, ξ_{7}. The 128 solutions are described above. The equations coming from Q are

ρ_{1}ξ_{1} + ρ_{2}ξ_{2} + ρ_{3}ξ_{3} + ρ_{4}ξ_{4} + ρ_{5}ξ_{5} + χ_{1} = 0
ρ_{1}ξ_{2} + ρ_{2}ξ_{3} + ρ_{3}ξ_{4} + ρ_{4}ξ_{5} + ρ_{5}ξ_{6} + χ_{2} = 0
ρ_{1}ξ_{3} + ρ_{2}ξ_{4} + ρ_{3}ξ_{5} + ρ_{4}ξ_{6} + ρ_{5}ξ_{7} + χ_{3} = 0

We view this as inhomogeneous linear equations for the ρ_{i}. The coefficient matrix

M = ( \begin{matrix}ξ_{1} & ξ_{2} & ξ_{3} & ξ_{4} & ξ_{5} \\ ξ_{2} & ξ_{3} & ξ_{4} & ξ_{5} & ξ_{6} \\ ξ_{3} & ξ_{4} & ξ_{5} & ξ_{6} & ξ_{7} \end{matrix} )

is the transpose of the coefficient matrix of the equations p_{2,0}ξ_{i} + p_{2,1}ξ_{i+1} + p_{2,2}ξ_{i+2} = 0, viewed as equations for the coefficients of p_{2}. If for a given solution of the equations π_{m} the matrix M has not full rank, then there exists a noncomposed pencil admitting the same solution. But then also p_{2,0}χ_{1} + p_{2,1}χ_{2} + p_{2,2}χ_{3} = 0, an equation which in general is not satisfied. We have 8 solutions which lie on a rational normal curve and 28 solutions on the secant variety of this curve. The equations of the secant variety are the maximal minors of M. Only for 91 solutions the matrix M has full rank. □

In the general case we get components of dimension 3 + 1 (the y-rolling factors deformation does not enter the equations), for solutions not on the rational curve, but on its secant variety the component has dimension 5, while we get a 6-dimensional component if p_{8} and q_{12} have a common root. This does not contradict the fact that all smoothing components of Gorenstein surface singularities have the same dimension, because we here only look at the restriction to negative degree.

Proposition 4.21. The general hyperplane section of a K3 surface of type (5, 4, 3, 2; 8, 4) or (4, 4, 4, 2; 8, 4) is a tetragonal curve of type (5, 5, 4; 8, 4), which lies on a rational surface with two double points.

Proof . We use coordinate transformations on the scroll to bring the hyperplane section into a normal form, while we suppose the coefficients of the equations to be general. Let as usual (X, Y, Z, W; s, t) be coordinates on the scroll. Let the hyperplane section be a_{0}W_{0} + a_{1}W_{1} + a_{2}W_{2} + ⋯ = (a_{0}s^{2} + a_{1}st + a_{2}t^{2})W + · · ·. By a transformation in (s, t) we achieve that a_{0} = a_{2} = 0, so the equation is W_{1} + · · ·. First consider the case (4, 4, 4, 2). By a suitable transformation w ↦ W +a_{2}(s, t)X+b_{2}(s, t)Y +c_{2}(s, t)Z we remove all terms with index 1, 2 or 3, leaving W_{1}+a_{0}X_{0}+b_{0}Y_{0}+c_{0}Z_{0}+a_{4}X_{4}+b_{4}Y_{4}+c_{4}Z_{4}. Taking a_{0}X + b_{0}Y + c_{0}Z as new X and a_{4}X + b_{4}Y + c_{4}Z as new Y brings us finally to X_{4}+W_{1}+Y_{0}. With coordinates (x, y, z; s, t) for the scroll S_{5,5,4} we get the hyperplane section by setting Z = z, X = sx, Y = ty and W = −t^{3}x−s^{3}y. The equation P does not involve the variable W so we have quadratic singularities if sx = ty = z = 0, which gives the points s = y = z = 0 and t = x = z = 0. In the case (5, 4, 3, 2) we can achieve W_{1} + Z_{0} + Z_{3} and we get the curve by X = x, Y = z, Z = sty and W = −(s^{3} + t^{3})y. The equation P: p_{2}X^{2} + p_{1}XY + p_{0}Y^{2} + XZ now gives p_{2}x^{2} + p_{1}xz + p_{0}z^{2} + stxy, which for general p_{i} has singular points at x = z = st = 0. □

To investigate the sufficiency of these conditions we look at the general cone of type (5, 5, 4; 8, 4). We may suppose that P has the form z^{2} + P_{2}(x, y). The equation P_{2}(x, y) describes a curve of type (2, 2) on S_{5,5} ≅ ℙ^{1}×ℙ^{1}. If this curve has a singular point, we may assume that it lies in the point x = s = 0. Under the assumption that the coefficient of stxy does not vanish we can transform the equation into the form (as^{2} + bt^{2})x^{2} + 2stxy + cs^{2}y^{2} and unfolding the singularity we get the equation

P_{2} = (as^{2} + bt^{2})x^{2} + 2stxy + (cs^{2} + dt^{2})y^{2} .

One can then write out the lifting conditions and base equations coming from the equation P. The result is that they have only trivial solutions if and only if abcd((ad + bc − 1)^{2} − 4abcd) ≠ 0, if and only if the curve P_{2} is nonsingular. If a singularity is present we assume it to be in x = s = 0, so d = 0. The equation Q gives three base equations, in which 2+2 pure rolling factors variables can enter. We analyze what happens if there is a second singularity. For b = d = 0 the equation P_{2} is divisible by s, and we do not find extensions. In case a = d = 0 the curve P_{2} splits into two curves of type (1, 1); we get two components with deformed scroll S_{4,4,4,2}. For c = d = 0 we have intersection of a line with a curve of type (2, 1) and we find two components with deformed scroll S_{5,4,3,2}.

Remark 4.22. For the general tetragonal cone with large g we found dim T^{1}(−1) = 9, but all deformations are obstructed. For special curves extensions may exist; also the dimension can be higher. Both conditions seem to be independent. As the number of base equations we find is always g − 7, having more variables increases the chances of finding solutions. In the borderline case studied above this may suffice to force the existence, but in general it does not. On the other, taking a general hyperplane section of a general tetragonal K3 surface will give a cone with dim T^{1}(−1) = 9. It would be interesting to find a property of a canonical curve which gives a sufficient condition for the existence of an extension.

Acknowledgement. I thank Jim Brawner for sending me his preprint. The computer computations were done with the program Macaulay [Bayer– Stillman].

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Jan Stevens Matematik Göteborgs universitet Chalmers tekniska högskola SE 412 96 Göteborg Sweden stevens@math.chalmers.se