Welschinger invariants of real del Pezzo surfaces of degree $\ge 2$

Abstract

We compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree at least 2. We show that under some conditions, for such a surface $X$ and a real nef and big divisor class $D$, through any generic collection of $-D{K}_X-1$ real points lying on a connected component of the real part of $X$ one can trace a real rational curve $C\in |D|$. This is derived from the positivity of appropriate Welschinger invariants. We furthermore show that these invariants are asymptotically equivalent, in the logarithmic scale, to the corresponding genus zero Gromov-Witten invariants. Our approach consists in a conversion of Shoval-Shustin recursive formulas counting complex curves on the plane blown up at seven points and of Vakil's extension of the Abramovich-Bertram formula for Gromov-Witten nvariants into formulas computing real enumerative invariants.

67 pages, 1 figure; as compared to the published version, the missing factor 2^m is inserted into the second sum of the right-hand side of formula (31) in Theorem 3.2(3)

References in corpus (1)

• The open Gromov-Witten-Welschinger theory of blowups of the projective plane

Cited by in corpus (13)

• The open Gromov-Witten-Welschinger theory of blowups of the projective plane
• On higher genus Welschinger invariants of Del Pezzo surfaces
• Real algebraic curves with large finite number of real points
• Real algebraic curves on real del Pezzo surfaces
• Steenrod Pseudocycles, Lifted Cobordisms, and Solomon's Relations for Welschinger's Invariants
• WDVV-Type Relations for Welschinger's Invariants: Applications
• Welschinger invariants of Blow-ups of symplectic 4-manifolds
• Qualitative aspects of counting real rational curves on real K3 surfaces
• Higher genus Welschinger invariants under real surgeries
• A remark on Gromov-Witten-Welschinger invariants of $\mathbb{C}{P}^3\#{\overline{\mathbb{C}P}}^3$
• The real Mordell-Weil group of rational elliptic surfaces and real lines on del Pezzo surfaces of degree $K^2=1$
• On wall-crossing invariance of certain sums of Welschinger numbers
• Combined count of real rational curves of canonical degree 2 on real del Pezzo surfaces with $K^2=1$