Refined enumerative invariants and mixed Welschinger invariants

Abstract

For real toric surfaces and conjugation invariant point conditions with all conjugate pairs on the boundary divisors, we prove that the signed count of real curves of arbitrary genus in the linear system through the given points is invariant under variation of the points, provided the reduced tropicalization is in general position. The proof is based on a new relative refined tropical invariant, which is invariant under variation of the point conditions and specializes at $y\to -1$ to this signed count; at $y\to 1$ the same invariant recovers the count of complex curves with prescribed tangency to the boundary. We extend the invariant to allow arbitrary tangency orders along the boundary and identify its $y\to 1$ limit with the corresponding complex count. Finally, we show that in positive genus the signed real count is not invariant when conjugate pairs are allowed in the interior, even under strong genericity assumptions.