# Affine geometry of strata of differentials

## TL;DR

This paper investigates the affine properties of strata of k-differentials on smooth curves, showing certain conditions prevent the existence of complete curves and exploring the affine nature of related invariant manifolds.

## Contribution

It establishes conditions under which strata of k-differentials are affine and confirms the affine property for specific Teichmüller invariant manifolds.

## Key findings

- Strata with prescribed poles of order at least k contain no complete curves.
- Affine invariant manifolds from Teichmüller dynamics are affine in certain cases.
- Confirmed affine properties for Teichmüller curves, Hurwitz spaces, hyperelliptic strata, and some low genus strata.

## Abstract

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper we study affine related properties of strata of $k$-differentials on smooth curves which parameterize sections of the $k$-th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least $k$, then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichm\"uller dynamics are affine varieties, and confirm the answer for Teichm\"uller curves, Hurwitz spaces of torus coverings, hyperelliptic strata as well as some low genus strata.