Linear systems over P1xP1 with base points of multiplicity bounded by three

TL;DR

This paper introduces a combinatorial approach to determine non-specialty of linear systems of curves with multiple points on P1xP1, leading to a classification of systems with multiplicities up to three.

Contribution

It presents a novel combinatorial method for proving non-specialty and classifies special linear systems on P1xP1 with bounded multiplicities.

Findings

Classified all special linear systems on P1xP1 with multiplicities ≤ 3.

Developed a combinatorial technique for non-specialty proofs.

Provided a systematic approach to analyze linear systems with multiple points.

Abstract

We propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general positions. As an application we obtain a classification of special linear systems on P1xP1 for which the multiplicities do not exceed 3.