New methods for determining speciality of linear systems based at fat points in P^n

TL;DR

This paper introduces new techniques for calculating the dimension of linear systems with fat points in projective space, extending existing methods and applying them to bounds on Seshadri constants and Nagata's Conjecture.

Contribution

Develops novel methods for analyzing linear systems with fat points in P^n, extending previous approaches and applying them to important conjectures and bounds.

Findings

New lower bounds on multi-point Seshadri constants in P^2

A new proof of cases of Iarrobino's analogue to Nagata's Conjecture in higher dimensions

Extension of existing techniques for linear systems with fat points

Abstract

In this paper we develop techniques for determining the dimension of linear systems of divisors based at a collection of general fat points in P^n by partitioning the monomial basis for the vector space of global sections of O(d). The methods we develop can be viewed as extensions of those developed by Dumnicki. We apply these techniques to produce new lower bounds on multi-point Seshadri constants of P^2 and to provide a new proof of a known result confirming the perfect-power cases of Iarrobino's analogue to Nagata's Conjecture in higher dimension.