On the Picard group of moduli spaces

TL;DR

This paper investigates the Picard groups of moduli spaces in positive characteristic, providing a $p$-adic proof of their structure and exploring related fundamental group schemes.

Contribution

It offers a novel $p$-adic proof that the Picard group of certain moduli spaces is isomorphic to the integers, and proves triviality of local fundamental group schemes for specific varieties.

Findings

Picard group of moduli of vector bundles is isomorphic to integers

Local fundamental group scheme of certain varieties is trivial

Provides a $p$-adic proof approach

Abstract

We study the Picard groups of moduli spaces in positive characteristics and we give a "$p$-adic" proof that the Picard group of moduli of vector bundles of fixed determinant is isomorphic to the group of integers. Along the way we prove that the local fundamental group scheme of a normal unirational projective variety is trivial. This is reminiscent of results of Serre and Nygaard who studied the fundamental groups of smooth, projective, unirational varieties.