Lattice polygons and families of curves on rational surfaces

TL;DR

This paper develops methods to identify minimal degree families of curves on rational surfaces, including toric surfaces, by translating geometric problems into lattice geometry and extending the approach to more general rational surfaces.

Contribution

It introduces a lattice geometry approach to find minimal degree families on toric and rational surfaces, broadening the scope of previous methods.

Findings

Reduced the problem to lattice geometry for toric surfaces

Extended the method to general rational complex projective surfaces

Provided a systematic way to find minimal degree families

Abstract

First we solve the problem of finding minimal degree families on toric surfaces by reducing it to lattice geometry. Then we describe how to find minimal degree families on, more generally, rational complex projective surfaces.