On singularity properties of convolutions of algebraic morphisms

TL;DR

This paper investigates the singularity properties of convolutions of algebraic morphisms over characteristic zero fields, demonstrating that repeated convolution improves smoothness and singularity properties, leading to flat morphisms with rational singularities.

Contribution

It establishes that sufficiently many convolutions of dominant morphisms yield flat morphisms with rational singularities, extending the understanding of singularity improvement via convolution in algebraic geometry.

Findings

Convolution powers become flat with rational singularities after a finite number of iterations.

The results hold uniformly for families of morphisms with bounded complexity.

Implications include good asymptotic behavior of fibers over finite fields.

Abstract

Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $V$ be a finite dimensional $K$-vector space. For two algebraic morphisms $\varphi:X\rightarrow V$ and $\psi:Y\rightarrow V$ we define a convolution operation, $\varphi*\psi:X\times Y\to V$, by $\varphi*\psi(x,y)=\varphi(x)+\psi(y)$. We then study the singularity properties of the resulting morphism, and show that as in the case of convolution in analysis, it has improved smoothness properties. Explicitly, we show that for any morphism $\varphi:X\rightarrow V$ which is dominant when restricted to each irreducible component of $X$, there exists $N\in\mathbb{N}$ such that for any $n>N$ the $n$-th convolution power $\varphi^{n}:=\varphi*\dots*\varphi$ is a flat morphism with reduced geometric fibers of rational singularities (this property is abbreviated (FRS)). By a theorem of Aizenbud and Avni, for $K=\mathbb{Q}$, this is equivalent to good asymptotic behavior of the size of the $\mathbb{Z}/p^{k}\mathbb{Z}$-fibers of $\varphi^{n}$ when ranging over both $p$ and $k$. More generally, we show that given a family of morphisms $\{\varphi_{i}:X_{i}\rightarrow V\}$ of complexity $D\in\mathbb{N}$ (i.e. that the number of variables and the degrees of the polynomials defining $X_{i}$ and $\varphi_{i}$ are bounded by $D$), there exists $N(D)\in\mathbb{N}$ such that for any $n>N(D)$, the morphism $\varphi_{1}*\dots*\varphi_{n}$ is (FRS).