Asymptotic behaviour of rational curves

TL;DR

This paper studies how the space of rational curve mappings to a variety behaves as the degree increases, focusing on the toric case and extending to general varieties using homogeneous coordinate rings.

Contribution

It provides a detailed analysis of the asymptotic properties of the moduli space of rational curves, especially highlighting the toric case and generalizations.

Findings

Asymptotic behavior characterized for toric varieties.

Extension of results to general varieties.

Use of homogeneous coordinate rings as a key tool.

Abstract

We investigate the asympotic behaviour of the moduli space of morphisms from the rational curve to a given variety when the degree becomes large. One of the crucial tools is the homogeneous coordinate ring of the variey. First we explain in details what happens in the toric case. Then we examine the general case.