# Picard group of moduli of curves of low genus in positive characteristic

## TL;DR

This paper computes the Picard group of the moduli stack of smooth curves of genus 3 to 5 in positive characteristic, using equivariant intersection theory and Chow ring relations.

## Contribution

It provides explicit calculations of the Picard group for low genus moduli stacks, advancing understanding of their geometric and algebraic structure.

## Key findings

- Picard group of moduli stacks for g=3 to 5 computed
- Cycle classes of certain divisors on al M_g determined
- Relations in the Chow ring of moduli stacks established

## Abstract

We compute the Picard group of the moduli stack of smooth curves of genus $g$ for $3\leq g\leq 5$, using methods of equivariant intersection theory. We base our proof on the computation of some relations in the integral Chow ring of certain moduli stacks of smooth complete intersections. As a byproduct, we compute the cycle classes of some divisors on $\mathcal{M}_g$.