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Floor diagrams relative to a conic, and GW-W invariants of Del Pezzo surfaces. (English) Zbl 1316.14104

This outstanding paper resolves two important enumerative problems. One principal result of the author is a recursive formula for computing Gromov-Witten invariants of any genus on arbitrary del Pezzo surfaces, i.e., for the counting of complex curves of any degree and genus on arbitrary del Pezzo surfaces. Another equally interesting result is a series of recursive formulas for the computation of Welschinger invariants of many real del Pezzo surfaces, in particular, surfaces of degree one, which were not accessible by previously used methods. These formulas are similar to the so-called floor-diagram formulas in tropical geometry introduced by the author jointly with G. Mikhalkin [in: Proceedings of the 15th Gökova geometry-topology conference, Gökova, Turkey, May 26–31, 2008. Cambridge, MA: International Press. 64–90 (2009; Zbl 1200.14106)]. The main ideas can be traced back to the work by L. Caporaso and J. Harris [Invent. Math. 131, No. 2, 345–392 (1998; Zbl 0934.14040)] and the version of the symplectic sum formula due to J. Li [J. Differ. Geom. 60, No. 2, 199–293 (2002; Zbl 1063.14069)]. The geometric background of the new formulas is a degeneration of a given real or complex rational surface into the union of ruled surfaces and the respective degeneration of the counted curves into certain reducible curves. Besides computation of several previously unknown Gromov-Witten and Welschinger invariants, the author indicates potential applications, among them criteria for positivity of certain Welschinger invariants.

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14P05 Real algebraic sets
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14P25 Topology of real algebraic varieties
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