An asymptotic Alexander-Hirschowitz theorem for surfaces

TL;DR

This paper proves that for large degrees, general double points impose the expected conditions on linear systems of surfaces, extending classical results and conjecturing similar behavior in higher dimensions.

Contribution

It establishes an asymptotic theorem for surfaces, showing expected conditions are imposed by double points, and conjectures this extends to higher-dimensional varieties.

Findings

Double points impose expected conditions for large degrees

The space of sections with prescribed singularities has expected dimension

Supports conjecture for higher-dimensional varieties

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.