Welschinger Invariants of Toric Del Pezzo Surfaces with Nonstandard Real Structures

The Welschinger invariants of real rational algebraic surfaces are natural analogs of the Gromov–Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces equipped with a nonstandard real structure. Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric structure) was established by Mikhalkin and the author. As a consequence we prove that for any real ample divisor $D$ on a surface $\Sigma$ under consideration, through any generic configuration of $c_1(\Sigma )D-1$ generic real points, there passes a real rational curve belonging to the linear system $|D|$.