JournalspmVol. 77, No. 1pp. 73–110

Motivic volumes of fibers of tropicalization

  • Jeremy Usatine

    Brown University, Providence, USA
Motivic volumes of fibers of tropicalization cover

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Abstract

Let TT be an algebraic torus over an algebraically closed field, let XX be a smooth closed subvariety of a TT-toric variety such that U=XTU = X \cap T is not empty, and let L(X)\mathscr{L}(X) be the arc scheme of XX. We consider a tropicalization map on L(X)L(XU)\mathscr{L}(X) \setminus \mathscr{L}(X \setminus U), the set of arcs of XX that do not factor through XUX \setminus U. We show that each fiber of this tropicalization map is a constructible subset of L(X)\mathscr{L}(X) and therefore has a motivic volume. We prove that if UU has a compactification with simple normal crossing boundary, then the generating function for these motivic volumes is rational, and we express this rational function in terms of certain lattice maps constructed in Hacking, Keel, and Tevelev's theory of geometric tropicalization. We explain how this result, in particular, gives a formula for Denef and Loeser's motivic zeta function of a polynomial. To further understand this formula, we also determine precisely which lattice maps arise in the construction of geometric tropicalization.

Cite this article

Jeremy Usatine, Motivic volumes of fibers of tropicalization. Port. Math. 77 (2020), no. 1, pp. 73–110

DOI 10.4171/PM/2045