Compactified moduli spaces of rational curves in projective homogeneous varieties

TL;DR

This paper compares different compactifications of the space of rational curves in projective homogeneous varieties, specifically for degrees up to 3, and computes Betti numbers for Grassmannians.

Contribution

It provides explicit blow-up and blow-down comparisons of compactifications and calculates Betti numbers for these spaces in Grassmannian cases.

Findings

Explicit comparison of compactifications via blow-ups and blow-downs.

Betti numbers computed for compactified moduli spaces in Grassmannians.

Applicable to degrees up to 3 in projective homogeneous varieties.

Abstract

The space of smooth rational curves of degree $d$ in a projective variety $X$ has compactifications by taking closures in the Hilbert scheme, the moduli space of stable sheaves or the moduli space of stable maps respectively. In this paper we compare these compactifications by explicit blow-ups and -downs when $X$ is a projective homogeneous variety and $d\leq 3$. Using the comparison result, we calculate the Betti numbers of the compactifications when $X$ is a Grassmannian variety.