Volume 22, issue 6 (2018)

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Tropical refined curve counting via motivic integration

Johannes Nicaise, Sam Payne and Franziska Schroeter

Geometry & Topology 22 (2018) 3175–3234

We propose a geometric interpretation of Block and Göttsche’s refined tropical curve counting invariants in terms of virtual χy specializations of motivic measures of semialgebraic sets in relative Hilbert schemes. We prove that this interpretation is correct for linear series of genus 1, and in arbitrary genus after specializing from χy –genus to Euler characteristic.

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Refined enumerative geometry, tropical geometry, motivic integration
Mathematical Subject Classification 2010
Primary: 14E18, 14G22, 14T05
Received: 7 April 2016
Revised: 6 February 2018
Accepted: 27 March 2018
Published: 23 September 2018
Proposed: Lothar Göttsche
Seconded: Dan Abramovich, Richard Thomas
Johannes Nicaise
Department of Mathematics
Imperial College
South Kensington Campus
United Kingdom
Department of Mathematics
KU Leuven
Leuven, Belgium
Sam Payne
Department of Mathematics
University of Texas at Austin
Austin, TX
United States
Franziska Schroeter
Fachbereich Mathematik
Universität Hamburg