A Caporaso–Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces

Abstract

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.

Recursive Formulas for Welschinger Invariants of the Projective Plane

“…However the number obtained is no longer an invariant (see [Wel05a] or [IKS09]). See section 7.2 for this discussion in the tropical setting.…”

“…If one plugs r = 0 in Theorem 5.2, then one finds the Itenberg-Kharlamov-Shustin formula from [IKS09].…”

“…However, one sees easily that they do not correspond to any invariant, even tropical. Worse, we give an example in section 7.2 which shows that there is no hope to define as in [IKS09] tropical relative Welschinger invariants for real configurations of points containing some pairs of complex conjugated points.…”

“…GathmannMarkwig tropical formula in the complex case and Itenberg-Kharlamov-Shustin's one in the totally real case are easily translated into the language of floor diagrams. Moreover, the quite simple combinatoric of these objects allows one to generalize easily the formula in [IKS09] to Welschinger invariants for collections of points mixing real points and pairs of complex conjugated ones.…”

“…Then, Itenberg, Kharlamov and Shustin adapted in [IKS09] this tropical approach to a real setting and produced a Caporaso-Harris like recursive formula for Welschinger invariants in the totally real case. Their formula involves not only Welschinger invariants, but also other numbers which reveal to be tropical relative Welschinger invariants.…”