Refined curve counting on complex surfaces
By L. Gottsche, Vivek V. Shende. 2012
73 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/780ce61001f1d5c8c8257bfc1814b9002bd3c230
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In Computational Theory and Mathematics
- I. Itenberg
- G. Mikhalkin
- I. Cherednik
- Eugenii Shustin
- Erwan Brugallé
- Eric Katz
- Melody Chan
- O. Y. Viro
In Mathematical Physics
- I. Itenberg
- G. Mikhalkin
- I. Danilenko
- I. Cherednik
- Eugenii Shustin
- Brett Parker
- Peter Samuelson
- H. Morton
Citations in the corpus
- DAHA and iterated torus knots (2014, background): “Versal deformations Last, but not least, we refer the reader to [24] for some relatively recent developments concerning the Hilbert schemes of (locally) plane curves, including their role in the theory of the Gopakumar–Vafa BPS invariants (see [35]) and various generalizations.”
- On Block-Goettsche multiplicities for planar tropical curves (2012, background): “lation of the refined Severi degrees (introduced by Göttsche in connection with [12]) that interpolate between the numbers of complex and real curves, see [8].”