Irreducible components of the global nilpotent cone

TL;DR

This paper provides a combinatorial description of irreducible components of the global nilpotent cone's semistable locus in genus ≥2, revealing their structure via twisted Jordan strata and polytopes.

Contribution

It introduces a combinatorial framework for understanding the irreducible components and semistability conditions of the global nilpotent cone, independent of coprimality assumptions.

Findings

Irreducible components correspond to twisted Jordan strata.

Semistability characterized by combinatorial conditions on Jordan types.

Attracting cells are irreducible regardless of coprimality.

Abstract

This paper gives a combinatorial description of the set of irreducible components of the semistable locus of the global nilpotent cone, in genus $\ge2$. The first main result of this paper states that the set of irreducible components of the global nilpotent cone is given by the very natural decomposition in twisted Jordan strata, which are smooth. Then we move on to the semistable locus and obtain purely combinatorial conditions on the twisted Jordan type to be semistable. The proof uses an analogous result obtained in the context of moduli stacks of chains, and shows that semistability can be tested on the most `simple' subsheaves - the ones built with iterated kernels and images. The proof is constructive and do not rely on the coprimality of the rank and degree, in particular we get that the attracting cells are irreducible in any case. The last section describes this set of semistable components in terms of integral polytopes.