#### Volume 18, issue 4 (2014)

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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
##### 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