An asymptotic Alexander-Hirschowitz theorem for surfaces

Abstract

Let X be a smooth projective surface over C and let L be an ample line bundle on X. In this note, we show that, for all sufficiently large d, any number of general double points on X imposes the expected number of conditions on the linear system |L^d|. Equivalently, the space of d-plane sections of X singular at any number of general points has the expected dimension. We conjecture that the same holds for X of arbitrary dimension.

It was pointed out to the author soon after posting that a result subsuming both the main theorem and conjecture of this paper were proven in: J. Alexander and A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, Invent. Math. 140 (2000), 303-325. This article is no longer intended for publication