Critical loci and second-order singularities in arbitrary characteristic

TL;DR

This paper studies the structure and singularities of critical loci of maps between smooth schemes over a field, revealing characteristic-dependent behaviors and providing local descriptions akin to Morse theory.

Contribution

It establishes smoothness properties of critical loci for general maps, computes codimensions of second-order singularities, and offers local descriptions at critical points in arbitrary characteristic.

Findings

Critical loci are smooth in characteristic not 2.

First critical locus may be singular at finitely many points in characteristic 2.

Codimensions of second order singularities match known topological results in characteristic not 2.

Abstract

The critical loci of a map $f:X\to Y$ between smooth schemes over a field $k$ are the locally closed subschemes $\Sigma^i(f)\subseteq X$ where the differential of $f$ has constant rank. We prove that if $f : X\to \mathbb A^r$ is the general member of a suitably large linear family of maps from a smooth $k$-scheme $X$ to affine space, then the critical loci $\Sigma^i(f)$ are smooth, except in characteristic 2 where the first critical locus $\Sigma^1(f)$ may be singular at a finite set of points. Moreover, we compute the codimensions of the loci of second order singularities of such general maps $f :X \to \mathbb A^r$. In characteristics different from 2, the codimensions we find agree with those found by Levine in the context of differential topology. Finally, assuming that $k$ is an algebraically closed and $\dim X\ge \dim Y$, we give a local description of an arbitrary map $f :X \to Y$ at points of its first critical locus $\Sigma^1(f)$. In the case of functions and nondegenerate critical points, this description recovers the usual one from Morse theory.