Some Remarks on Morphisms

Between Fano Threefolds:

Erratum

cf. Documenta Math. 9, p. 471–486

Ekaterina Amerik

Received: July 5, 2005

Communicated by Thomas Peternell

Abstract. It was pointed out to me by James McKernan that there is a mistake in the proof of Proposition 2.7. Namely, it is stated (and “used”) there that the complement to a smooth conic in ℙ^{2} is simplyconnected. This is obviously false; indeed, the fundamental group is cyclic of order two. This mistake is, however, easy to deal with.

2000 Mathematics Subject Classification: 14J45

Keywords and Phrases: Fano threefolds, connectedness

It was pointed out to me by James McKernan that there is a mistake in the proof of Proposition 2.7. Namely, it is stated (and “used”) there that the complement to a smooth conic in ℙ^{2} is simply-connected. This is obviously false; indeed, the fundamental group is cyclic of order two. This mistake is, however, easy to deal with. What I propose below is the corrected version of the end of the proof of Proposition 2.7, starting from the line 20. Instead of “But the latter is simply-connected...” and so forth, read: “Remark that the fundamental group of the latter is cyclic of order two. In the case deg(X) = 4 and m = 5, the degree of p is 10; this means that H has at least 5 irreducible components. Each of them gives at least one conic through a general point of X. These conics are mapped to different lines on V_{5}, because they intersect. This is a contradiction because on V_{5} there are only three lines through a general point.

In the other case deg(X) = 10 and m = 2, the degree of p is 4, and this means that H cannot be irreducible. To derive a contradiction, we need another easy observation, which is made for example in [HM] (Proposition 12). It says that (thanks to the fact that P ic(X) = ℤ and that the normal bundle of a general conic on X is trivial) for H_{0} an irreducible component of H and C_{0} the universal family of conics over H_{0}, the evaluation map C_{0} → X cannot be generically one-to-one. So in fact each irreducible component of H provides at least two conics through a general point of X and therefore at least two lines through a general point of V_{5}; H must thus be irreducible, a contradiction.”

References

[HM] J.M. Hwang, N. Mok, Finite morphisms onto Fano manifolds of Picard number 1 which have rational curves with trivial normal bundles, J. Algebraic Geom. 12 (2003), no. 4, 627–651.

Ekaterina Amerik Université Paris-Sud Laboratoire des Mathématiques Bâtiment 425 91405 Orsay France Ekaterina.Amerik@math.u-psud.fr