Welschinger invariants of real Del Pezzo surfaces of degree ≥ 3

Abstract

We give a recursive formula for purely real Welschinger invariants of real Del Pezzo surfaces of degree K 2 ≥ 3, where in the case of surfaces of degree 3 with two real components we introduce a certain modification of Welschinger invariants and enumerate exclusively the curves traced on the non-orientable component. As an application, we prove the positivity of the invariants under consideration and their logarithmic asymptotic equivalence, as well as congruence modulo 4, to genus zero Gromov-Witten invariants.

Floor diagrams relative to a conic, and GW–W invariants of Del Pezzo surfaces

“…The sign of Welschinger invariants seem to obey to some mysterious rule related to the topology of the real part of the ambient manifold. The present work together with [Wel07], [IKS09], [IKS13b], [BP13], and [BP14] explicit this rule in a few cases, namely when L = T 2 , or S 2 and r = 0, 1, when X = X 8 and s is very small, or when L intersects a real Lagrangian sphere in a single point and r = 1. In the particular case of Del Pezzo surfaces, floor diagrams relative to a conic, with either empty or non-empty real part, provide a unified way to prove this rule when either r or s is small.…”

“…Results about the sign of Welschinger invariants, their sharpness, their arithmetical properties, their vanishing, and comparison of real and complex invariants were for example previously obtained in this way in [IKS03,IKS04,IKS09,IKS13c,IKS13b,IKS13a,Wel07,BP13,BP14]. Several extensions of those results are deduced from the methods presented here, see Corollaries 4.4, 4.5, 6.10, 6.11, 7.6, 7.7, 7.8, and Proposition 8.1.…”

“…does not depend on x as long as |Rx| = r, neither on the deformation class of (X n , c) [Wel03,Wel05a,IKS13b]. Those numbers are known as Welschinger invariants of (X n , c).…”

Relative enumerative invariants of real nodal del Pezzo surfaces

“…. , E 8 in Pic(Σ) as in (11) under the numbering of points p i in a way that p 1 , p 2 do not belong neither to E ′ or E 0 (in notation of Lemmas 2.6 and 2.8), and the surface Σ ′ obtained by contracting σ : Σ → Σ ′ of E 1 , E 2 . If | − K Σ | contains a cuspidal curve C for any position of p 1 , p 2 , we obtain at least a twodimensional family of cuspidal curves…”

Welschinger invariants of blow-ups of symplectic 4-manifolds

“…Given J ∈ RJ ω , the set RC(d, x, J) consists of real rational J-holomorphic curves f : S → X in X realizing the class d, passing through x, and such that f (RS) ⊂ L. Note that if r 1, the condition f (RS) ⊂ L is always satisfied. Itenberg, Kharlamov and Shustin [20] observed that the integer…”