Non-reduced components of global nilpotent cones

TL;DR

This paper characterizes the non-reduced components of global nilpotent cones across various geometric contexts, revealing they are generally nowhere reduced under certain conditions.

Contribution

It provides new results on the non-reduced structure of global nilpotent cones in Hitchin fibrations and moduli spaces on surfaces, using advanced geometric techniques.

Findings

Global nilpotent cone for Hitchin fibration is nowhere reduced under certain conditions.

Moduli space of one-dimensional sheaves on surfaces has nowhere reduced global nilpotent cone.

General fiber of Beauville-Mukai system has primitive homology class only when r=1.

Abstract

We determine the non-reduced components of global nilpotent cones in various cases of interest. In particular, under the appropriate coprimality conditions, we show: (1) the global nilpotent cone for an $L$-twisted $\operatorname{GL}_r$-Hitchin fibration associated to a curve $C$ of genus $g\ge 2$ is nowhere reduced, where $L$ is either the canonical bundle or has degree greater than $2g-2$; (2) the global nilpotent cone for a moduli space of one-dimensional sheaves on a K3, abelian, or del Pezzo surface is nowhere reduced; (3) suppose $\ell$ is a primitive, basepoint-free, big and nef class on a K3 surface, then a general fiber of a Beauville-Mukai system for the class $r\ell$ has primitive homology class if and only if $r=1$. Our methods include group scheme actions on Lagrangian fibrations, a GIT-stratification of global nilpotent cones of Hitchin fibrations, and deformation to the normal cone.