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Welschinger invariants of small non-toric del Pezzo surfaces. (English) Zbl 1307.14073

The authors have studied Welschinger invariants of real toric Del Pezzo surfaces using tropical methods and a Caporaso-Harris degeneration technique [I. Itenberg et al., Comment. Math. Helv. 84, No. 1, 87–126 (2009; Zbl 1184.14092)]. In this work, they consider non-toric Del Pezzo surfaces. Since tropical methods are in principle limited to the toric situation, they have to find a way to overcome this problem. They still rely on tropical methods by blowing down exceptional divisors and carefully studying the relation of the curves to count. The authors derive interesting applications, e.g., results about the asymptotic behaviour of Welschinger invariants when compared to Gromov-Witten invariants.

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14P05 Real algebraic sets
14T05 Tropical geometry (MSC2010)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)

Citations:

Zbl 1184.14092
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References:

[1] Arroyo, A., Brugallé, E., Lopez de Medrano, L.: Recursive formulas for Welschinger in- variants of the projective plane. Int. Math. Res. Notices 2011, 1107-1134 · Zbl 1227.14047 · doi:10.1093/imrn/rnq096
[2] Brugallé, E., Mikhalkin, G.: Enumeration of curves via floor diagrams. C. R. Math. Acad. Sci. Paris 345, 329-334 (2007) · Zbl 1124.14047 · doi:10.1016/j.crma.2007.07.026
[3] Caporaso, L., Harris, J.: Counting plane curves of any genus. Invent. Math. 131, 345-392 (1998) · Zbl 0934.14040 · doi:10.1007/s002220050208
[4] Gathmann, A., Markwig, H.: The numbers of tropical plane curves through points in general position. J. Reine Angew. Math. 602, 155-177 (2007) · Zbl 1115.14049 · doi:10.1515/CRELLE.2007.006
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