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On singularity properties of convolutions of algebraic morphisms

  • Itay Glazer
  • Yotam I. HendelEmail author


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, \(\varphi *\psi :X\times Y\rightarrow V\), 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 nth 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).

Mathematics Subject Classification

03C98 14B05 14E18 11G25 14G05 



We thank Moshe Kamenski and Raf Cluckers for enlightening conversations about the model theoretic settings. We thank Nir Avni for numerous helpful discussions, as well as for proposing this problem together with Rami Aizenbud. A large part of this work was carried out while visiting the mathematics department at Northwestern university, we thank them and Nir for their hospitality. Finally we wish to thank our teacher Rami Aizenbud for answering various questions and for helping to shape many of the ideas in this paper. We benefited from his guidance deeply. We also wish to thank the anonymous referees for their insightful comments and remarks, and in particular for suggesting the alternative proof of Theorem 5.2 (see Sect. 5.2). Both authors where partially supported by ISF Grant 687/13, BSF Grant 2012247 and a Minerva Foundation Grant.


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Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceThe Weizmann Institute of ScienceRehovotIsrael

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