Vol. 15, No. 9, 2021

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Motivic Euler products in motivic statistics

Margaret Bilu and Sean Howe

Vol. 15 (2021), No. 9, 2195–2259
Abstract

We formulate and prove an analog of Poonen’s finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil and Wood, who treated the case of smooth hypersurface sections, and is made possible by the use of motivic Euler products to write down candidate motivic probabilities. As applications, we give motivic analogs of many results in arithmetic statistics that have been proven using Poonen’s sieve, including work of Bucur and Kedlaya on complete intersections and Erman and Wood on semiample Bertini theorems.

Keywords
Bertini problems, Grothendieck ring of varieties, motivic probability, radicial surjective morphisms
Mathematical Subject Classification 2010
Primary: 14G10
Secondary: 55R80
Milestones
Received: 3 December 2019
Revised: 1 February 2021
Accepted: 17 March 2021
Published: 23 December 2021
Authors
Margaret Bilu
Institute of Science and Technology Austria
Am Campus 1
Klosterneuburg
Austria
Sean Howe
Department of Mathematics
University of Utah
Salt Lake City, UT
United States