Enumeration of non-nodal real plane rational curves

TL;DR

This paper investigates the existence and uniqueness of real rational plane curves with prescribed non-nodal singularities, extending the concept of Welschinger invariants to these cases and demonstrating their application to three-cuspidal quartics.

Contribution

It introduces a new type of enumerative invariant for real rational curves with non-nodal singularities and proves its uniqueness and existence for three-cuspidal quartics.

Findings

The invariant counting three-cuspidal quartics passing through four pairs of conjugate points is unique.

Such quartics always exist for any generic configuration of four pairs of conjugate points.

The study extends Welschinger invariants to non-nodal singularities in real algebraic geometry.

Abstract

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. Consequently, we show that through any generic configuration of four pairs of complex conjugate points, one can always trace a pair of real rational three-cuspidal quartics.