On higher genus Welschinger invariants of Del Pezzo surfaces

Abstract

The Welschinger invariants of real rational algebraic surfaces count real rational curves which represent a given divisor class and pass through a generic conjugation-invariant configuration of points. No invariants counting real curves of positive genera are known in general. We indicate particular situations, when Welschinger-type invariants counting real curves of positive genera can be defined. We also prove the positivity and give asymptotic estimates for such Weischinger-type invariants for several del Pezzo surfaces of degree $\ge 2$ and suitable real nef and big divisor classes. In particular, this yields the existence of real curves of given genus and of given divisor class passing through any appropriate configuration of real points on the given surface.

arXiv comment29 pages, updated to published version, International Mathematics Research Notices 2014

References in corpus (5)

• Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
• Welschinger invariants of real del Pezzo surfaces of degree $\ge 2$
• The open Gromov-Witten-Welschinger theory of blowups of the projective plane
• Optimalité, congruences et calculs d'invariants des variétés symplectiques réelles de dimension quatre
• Logarithmic asymptotics of the genus zero Gromov-Witten invariants of the blown up plane

Cited by in corpus (4)

• Real algebraic curves on real del Pezzo surfaces
• Welschinger invariants of Blow-ups of symplectic 4-manifolds
• On the lower bounds for real double Hurwitz numbers
• Higher genus Welschinger invariants under real surgeries