On singularity properties of convolutions of algebraic morphisms

• Itay Glazer ^[1] ; Yotam I. Hendel ^[1]
  1. [1] Weizmann Institute of Science

     Weizmann Institute of Science

     Israel

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 25, Nº. 1, 2019
• Idioma: inglés
• DOI: 10.1007/s00029-019-0457-z
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• Resumen

  • 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→V and ψ:Y→V we define a convolution operation, φ∗ψ:X×Y→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→V which is dominant when restricted to each irreducible component of X, there exists N∈N such that for any n>N the nth convolution power φn:=φ∗⋯∗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=Q , this is equivalent to good asymptotic behavior of the size of the Z/pkZ -fibers of φn when ranging over both p and k. More generally, we show that given a family of morphisms {φi:Xi→V} of complexity D∈N (i.e. that the number of variables and the degrees of the polynomials defining Xi and φi are bounded by D), there exists N(D)∈N such that for any n>N(D) , the morphism φ1∗⋯∗φn is (FRS).