Simple obstructions and cone reduction

TL;DR

This paper proves that the intrinsic normal cone of a Deligne-Mumford stack is supported in a specific subcone, providing an alternative proof of cone reduction by cosections and discussing vanishing of simple obstructions via semiregularity maps.

Contribution

It demonstrates the support of the intrinsic normal cone in a subcone and offers a new proof of cone reduction, also exploring obstruction vanishing under semiregularity.

Findings

Support of $C_X$ in $V(Omega_X[-1])$ established.

Provides an alternative proof of cone reduction by cosections.

Discusses vanishing of simple obstructions via Buchweitz-Flenner semiregularity map.

Abstract

Let $X$ be a Deligne-Mumford stack locally of finite type over an algebraically closed field $k$ of characteristic zero. We show that the intrinsic normal cone $C_X$ of $X$ is supported in the subcone $\mathbb{V}(\Omega_X[-1])$ ($h^1/h^0((\Omega^1_X)^\vee)$) of its intrinsic normal sheaf $N_X$. This leads to an alternative proof of cone reduction by cosections for $C_X$. We also discuss vanishing of simple obstructions under the Buchweitz-Flenner semiregularity map for sheaves.