Limit linear systems and applications

TL;DR

This paper investigates the regularity of certain plane curve systems with prescribed multiplicities, establishing a condition for regularity based on the number of points, and explores the computation of limits of linear systems with fixed divisors.

Contribution

It provides a new criterion for the regularity of linear systems of plane curves with multiple points and studies the limits of these systems with fixed divisors.

Findings

Regularity when n > (2m)^2

Computation of limits of linear systems with fixed divisors

Insight into the behavior of linear systems with prescribed multiplicities

Abstract

A system of plane curves defined by prescribing n points of multiplicity m in general position is regular if n > (2m)^2. The proof uses computation of limits of linear systems acquiring fixed divisors, an interesting problem in itself.