We define a series of relative tropical Welschinger-type invariants of real toric surfaces. In the Del Pezzo case, these invariants can be seen as real tropical analogs of relative Gromov-Witten invariants, and are subject to a recursive formula. As application we obtain new formulas for Welschinger invariants of real toric Del Pezzo surfaces.
'est pas un livre, un texte qu'on lit du débutà la fin, mais un ouvrage que l'on consulte quand on en a besoin. Lorsqu'on hésite sur une voieà suivre, une attitudeà prendre, un choixà faire, un dilemmeà résoudre, on peut alors s'en servir pour ce qu'il est dans la pratique : un manuel d'aideà la décision.Cyrille Javary, Les Rouages du Yi Jing, Ed. Phillipe Picquier, 2001Abstract. We study real elliptic surfaces and trigonal curves (over a base of an arbitrary genus) and their equivariant deformations. We calculate the real TateShafarevich group and reduce the deformation classification to the combinatorics of a real version of Grothendieck's dessins d'enfants. As a consequence, we obtain an explicit description of the deformation classes of M -and (M − 1)-(i.e., maximal and submaximal in the sense of the Smith inequality) curves and surfaces.
International audienceWe compute the purely real Welschinger invariants, both original and modified, for all real del Pezzo surfaces of degree ≥ 2. We show that under some conditions, for such a surface X and a real nef and big divisor class D ∈ Pic(X), through any generic collection of −DK X − 1 real points lying on a connected component of the real part RX of X one can trace a real rational curve C ∈ |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 invariants into formulas computing real enumerative invariants
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.