Combinatorial Göttsche-Schroeter invariants in any genus

Göttsche-Schroeter invariants are a genus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>0</mml:mn> </mml:math> extension of Block-Göttsche invariants. They interpolate between Welschinger invariants involving pairs of complex conjugated points and genus <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>0</mml:mn> </mml:math> descendant Gromov-Witten invariants. They can be computed by a floor diagram algorithm. In this paper, we show that this floor diagrams recipe actually leads to some invariants in any genus. This generalizes Göttsche-Schroeter invariant in higher genus in a combinatorial way. We then prove some polynomiality result and establish a link with invariants defined by Shustin and Sinichkin. We provide many examples. In particular, we conjecture that these combinatorial invariants satisfy the Abramovich-Bertram formula.