# Uniruledness of Strata of Holomorphic Differentials in Small Genus

## TL;DR

This paper investigates the birational geometry of strata of holomorphic and quadratic differentials in small genus, demonstrating they are uniruled or unirational by constructing explicit rational curves and bundles.

## Contribution

It introduces new constructions of rational curves and projective bundles to prove uniruledness and unirationality of these strata in small genus.

## Key findings

- Strata are uniruled in small genus via rational curves from K3 and del Pezzo surfaces.
- Holomorphic strata are shown to be unirational for genus 3 to 6.
- Explicit projective bundles dominate the strata, establishing their birational properties.

## Abstract

We address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials. We show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on K3 and del Pezzo surfaces respectively. Restricting to genus $3\leq g\leq6$, we construct projective bundles over a rational varieties that dominate the holomorphic strata with length at most $g-1$, hence showing in addition that these strata are unirational.