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Welschinger invariants of toric del Pezzo surfaces with nonstandard real structures. (English) Zbl 1184.14083

Proc. Steklov Inst. Math. 258, 218-246 (2007) and Tr. Mat. Inst. Steklova 258, 227-255 (2007).
A real algebraic surface is a complex algebraic one equipped with an anti-holomorphic involution. A curve (respectively, a collection of points) on a real algebraic surface \((\Sigma, c)\) is called real, if the curve (respectively, the collection of points) is invariant under the anti-holomorphic involution \(c:\Sigma\to\Sigma\).
The Welschinger invariants can be seen as real analogs of genus zero Gromov-Witten invariants and are designed to bound from below the number of real rational curves passing through a given generic real collection of points on a real rational surface. In some cases (for example, for toric Del Pezzo surfaces equipped with the tautological real structure, i.e., the real structure naturally compatible with the toric structure), these invariants can be calculated using Mikhalkin’s approach which deals with a corresponding count of tropical curves [see G. Mikhalkin, J. Am. Math. Soc. 18, No. 2, 313–377 (2005; Zbl 1092.14068) and E. Shustin, J. Algebr. Geom. 15, No. 2, 285–322 (2006; Zbl 1118.14059)].
The paper under review contains a tropical formula for Welschinger invariants of four toric Del Pezzo surfaces equipped with a non-tautological real structure. The first surface is \({\mathbb P}^1 \times {\mathbb P}^1\) equipped with the real structure \((z, w) \mapsto ({\overline w}, {\overline z})\); the real point set of this surface is a sphere. The other three surfaces are obtained by blowing up the first one in, respectively, one real point, two real points, and two imaginary conjugated points. As a consequence, the author proves that, for any real ample divisor \(D\) on a surface \(\Sigma\) under consideration and for any generic configuration \(\omega\) of \(c_1(\Sigma) \cdot D - 1\) real points on \(\Sigma\), there always exists a real rational curve belonging to the linear system \(\mid D \mid\) and passing through all the points of \(\omega\).

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14T05 Tropical geometry (MSC2010)
14P25 Topology of real algebraic varieties
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