Algebraically primitive invariant subvarieties with quadratic field of definition

TL;DR

This paper classifies algebraically primitive invariant subvarieties of translation surface strata with quadratic fields, identifying specific known examples and their unique properties.

Contribution

It proves that only certain well-known invariant subvarieties are algebraically primitive with quadratic fields of definition.

Findings

Decagon, Weierstrass curves, and eigenform loci are the only such subvarieties.

Identifies a unique rank two example in genus four.

Provides a classification result for these invariant subvarieties.

Abstract

We show that the only algebraically primitive invariant subvarieties of strata of translation surfaces with quadratic field of definition are the decagon, Weierstrass curves, and eigenform loci in genus two and the rank two example in the minimal stratum of genus four translation surfaces discovered by Eskin-McMullen-Mukamel-Wright.