Elliptic surfaces and linear systems with fat points

TL;DR

This paper studies the dimension of linear systems with fat points on surfaces, using elliptic surface specialization, and explores characteristic-dependent behaviors with new conjectures and proofs.

Contribution

It introduces a specialization approach to elliptic surfaces for analyzing linear systems with fat points and proposes a conjecture valid in characteristic zero.

Findings

Special case of one fat point implies general case for multiple points in characteristic zero.

The conjecture fails in positive characteristic.

Results extend to other surfaces beyond elliptic surfaces.

Abstract

We investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces. For the projectivization of the Atiyah bundle over an elliptic curve with a certain polarization, we observe that the special case of only one fat point implies the general case of arbitrarily many fat points, as well as results concerning other surfaces. We conjecture that this special case holds in characteristic 0, but prove that it fails in any positive characteristic.