# Non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials

## TL;DR

This paper explores the structure of bi-algebraic curves and surfaces within moduli spaces of Abelian differentials, demonstrating the existence of non-linear examples that challenge previous linearity results under certain conditions.

## Contribution

It constructs explicit non-linear bi-algebraic curves and surfaces in moduli spaces of Abelian differentials, expanding understanding beyond the linear cases proven under condition $(ullet)$.

## Key findings

- Existence of non-linear bi-algebraic curves in genus 7
- Existence of non-linear bi-algebraic surfaces in genus 10
- Challenges previous linearity results under condition $(ullet)$

## Abstract

The strata of the moduli spaces of Abelian differentials are non-homogenous spaces carrying natural bi-algebraic structures. Partly inspired by the case of homogenous spaces carrying bi-algebraic structures (such as torii, Abelian varieties and Shimura varieties), Klingler and Lerer recently showed that any bi-algebraic curve in a stratum of the moduli space of Abelian differentials is linear provided that the so-called condition $(\star)$ is fulfilled. In this note, we construct a non-linear bi-algebraic curve, resp. surface, of Abelian differentials of genus $7$, resp. $10$.