Paper detail

On singularity properties of convolutions of algebraic morphisms

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 $φ:X\rightarrow V$ and $ψ:Y\rightarrow V$ we define a convolution operation, $φ*ψ:X\times Y\to V$, by $φ*ψ(x,y)=φ(x)+ψ(y)$. 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 $φ:X\rightarrow V$ 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 $φ^{n}:=φ*\dots*φ$ 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 $φ^{n}$ when ranging over both $p$ and $k$. More generally, we show that given a family of morphisms $\{φ_{i}:X_{i}\rightarrow V\}$ of complexity $D\in\mathbb{N}$ (i.e. that the number of variables and the degrees of the polynomials defining $X_{i}$ and $φ_{i}$ are bounded by $D$), there exists $N(D)\in\mathbb{N}$ such that for any $n>N(D)$, the morphism $φ_{1}*\dots*φ_{n}$ is (FRS).