On Welschinger invariants of descendant type

TL;DR

This paper introduces new enumerative invariants for real del Pezzo surfaces that count real rational curves with specified tangency conditions, demonstrating their invariance under various geometric choices and relating them to complex descendant invariants.

Contribution

It defines and proves invariance of real descendant-type Welschinger invariants for real del Pezzo surfaces, extending enumerative geometry tools to real algebraic geometry.

Findings

Invariants are independent of point-arc configuration

Invariants are unaffected by surface variations

They serve as real analogs of complex descendant invariants

Abstract

We introduce enumerative invariants of real del Pezzo surfaces that count real rational curves belonging to a given divisor class, passing through a generic conjugation-invariant configuration of points and satisfying preassigned tangency conditions to given smooth arcs centered at the fixed points. The counted curves are equipped with Welschinger-type signs. We prove that such a count does not depend neither on the choice of the point-arc configuration, nor on the variation of the ambient real surface. These invariants can be regarded as a real counterpart of (complex) descendant invariants.