Rationality and Chow-Kuenneth decompositions for some moduli stacks of curves

TL;DR

This paper establishes Chow-Kuenneth decompositions for certain moduli stacks of curves and explores the birational geometry and rationality of a specific moduli space of double covers of genus 3 curves.

Contribution

It proves the existence of Chow-Kuenneth decompositions for low-genus moduli stacks and constructs a birational model of the moduli space R as a group quotient, advancing understanding of their geometric structures.

Findings

Chow-Kuenneth decomposition exists for low-genus moduli stacks of curves.

A birational model of the moduli space R is constructed as a group quotient of Grassmannian products.

The rationality of the moduli space R is discussed.

Abstract

In this paper, we show the existence of a Chow--Kuenneth decomposition for the moduli stack of stable curves of genus g with r marked points, for low values of g,r. We also look at the moduli space R of double covers of genus 3 curves, branched along 4 distinct points. We obtain a birational model of the moduli space R as a group quotient of a product of two Grassmanian varieties. This provides a Chow-Kuenneth decomposition over an open subset of R. The question of rationality of R is also discussed.