On the vector bundles from Chang and Ran's proof of the unirationality of $\mathcal{M}_g$, $g \leq 13$

TL;DR

This paper offers an alternative proof for the unirationality of moduli spaces of low-genus curves by combining Chang and Ran's monad approach with a classification of certain vector bundles on projective space.

Contribution

It introduces a new method that merges monad techniques with vector bundle classification to establish unirationality for moduli spaces of curves up to degree 13.

Findings

Unirationality proven for moduli spaces of curves with degree ≤ 13

New classification results for globally generated vector bundles on projective 3-space

Alternative proof approach combining monads and vector bundle classification

Abstract

We combine the idea of Chang and Ran [Invent. Math. 76 (1984), 41-54] of using monads of vector bundles on the projective 3-space to prove the unirationality of the moduli spaces of curves of low genus with our classification of globally generated vector bundles with small first Chern class $c_1$ on the projective 3-space to get an alternative argument for the unirationality of the moduli spaces of curves of degree at most 13 (based on the general framework of Chang and Ran).