Volume 18, issue 4 (2014)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Refined curve counting on complex surfaces

Lothar Göttsche and Vivek Shende

Geometry & Topology 18 (2014) 2245–2307
Abstract

We define refined invariants which “count” nodal curves in sufficiently ample linear systems on surfaces, conjecture that their generating function is multiplicative, and conjecture explicit formulas in the case of K3 and abelian surfaces. We also give a refinement of the Caporaso–Harris recursion, and conjecture that it produces the same invariants in the sufficiently ample setting. The refined recursion specializes at y = 1 to the Itenberg–Kharlamov–Shustin recursion for Welschinger invariants. We find similar interactions between refined invariants of individual curves and real invariants of their versal families.

Keywords
Hilbert schemes of points, Severi degrees, Donaldson–Thomas invariants, Welschinger invariants
Mathematical Subject Classification 2010
Primary: 14C05, 14H20
Secondary: 14N10, 14N35
References
Publication
Received: 13 February 2013
Accepted: 8 March 2014
Published: 2 October 2014
Proposed: Richard Thomas
Seconded: Jim Bryan, Ronald Stern
Authors
Lothar Göttsche
International Centre for Theoretical Physics
Strada Costiera 11
34151 Trieste
Italy
http://users.ictp.it/~gottsche
Vivek Shende
Department of Mathematics
University of California, Berkeley
970 Evans Hall
Berkeley, CA 94720-3840
USA
http://math.berkeley.edu/~vivek