Real algebraic curves on real del Pezzo surfaces

TL;DR

This paper advances the understanding of the topology of real algebraic curves on non-toric surfaces, specifically real del Pezzo surfaces of degrees 1 and 2, using degeneration and enumerative geometry techniques.

Contribution

It provides new obstructions and constructions for the topological classification of real algebraic curves on certain non-toric surfaces.

Findings

Obstructions to topological types of real algebraic curves identified.

Constructed real algebraic curves with prescribed topology.

Extended classical methods to non-toric del Pezzo surfaces.

Abstract

The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves in real toric surfaces is a classical subject that has undergone considerable evolution. On the other hand, not much is known for more general ambient surfaces. We take a step forward in the study of topological types classification of real algebraic curves on non-toric surfaces focusing on real del Pezzo surfaces of degree $1$ and $2$ with multi-components real part. We use degeneration methods and real enumerative geometry in combination with variations of classical methods to give obstructions to topological types and to give constructions of real algebraic curves with prescribed topology.