Refined enumerative invariants and mixed Welschinger invariants

TL;DR

This paper introduces a new refined tropical invariant for real algebraic curves on toric surfaces, proving its invariance under certain conditions and relating it to complex curve counts, while also exploring limitations in positive genus cases.

Contribution

It develops a relative refined tropical invariant that unifies real and complex curve counts and proves its invariance under specific geometric conditions.

Findings

The invariant specializes to signed real counts at y→-1.

At y→1, it recovers complex curve counts.

In positive genus, the signed real count is not invariant with interior conjugate pairs.

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.