Refined curve counting on complex surfaces

Göttsche, Lothar¹ ; Shende, Vivek²

¹ International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy
² Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA 94720-3840, USA

Geometry & topology, Tome 18 (2014) no. 4, pp. 2245-2307.

Résumé

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.

Détail

DOI : 10.2140/gt.2014.18.2245

Classification : 14C05, 14H20, 14N10, 14N35
Keywords: Hilbert schemes of points, Severi degrees, Donaldson–Thomas invariants, Welschinger invariants

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