Two remarks on the Weierstrass flag

TL;DR

This paper investigates the geometric properties of Weierstrass flag strata on moduli spaces of curves, revealing that these strata are generally not affine, which impacts their geometric and algebraic understanding.

Contribution

It demonstrates that the locally closed strata of Weierstrass flags are almost never affine on moduli spaces of curves, providing new insights into their geometric structure.

Findings

Weierstrass flag strata are almost never affine.

The non-affineness impacts geometric and algebraic properties.

Results apply to both genus g and genus g with marked points.

Abstract

We show that the locally closed strata of the Weierstrass flags on the moduli spaces of curves of genus g and on the moduli space of curves of genus g with one marked point are almost never affine.