Welschinger invariants of toric Del Pezzo surfaces with non-standard real structures

DOI10.1134/S0081543807030157arXivmath/0605697MaRDI QIDQ6477263

Eugenii Shustin

Publication date: 27 May 2006

Abstract: The Welschinger invariants of real rational algebraic surfaces are natural analogues of the Gromov-Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces, equipped with a non-standard real structure. Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric structure) was established by Mikhalkin and the author. As a consequence we prove that, for any real ample divisor $D$ on a surfaces $Sigma$ under consideration, through any generic configuration of $c_1(Sigma)D-1$ generic real points there passes a real rational curve belonging to the linear system $|D|$.

Mathematics Subject Classification ID

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

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