Real Algebraic Curves on Real del Pezzo Surfaces

Abstract

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-type 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 the existence of topological-type classes realized by real algebraic curves and to give constructions of real algebraic curves with prescribed topology.