Real-fibered morphisms of del Pezzo surfaces and conic bundles

TL;DR

This paper characterizes real-fibered morphisms of real algebraic varieties, especially surfaces, using topological linking numbers, and classifies such morphisms for real del Pezzo surfaces and conic bundles.

Contribution

It provides a topological criterion for real-fibered morphisms and classifies all such morphisms from real del Pezzo surfaces to the projective plane.

Findings

Classified all real-fibered morphisms from real del Pezzo surfaces to the projective plane.

Developed a criterion based on linking numbers for real-fibered morphisms as linear projections.

Gave insights into real conic bundles and their morphisms.

Abstract

It goes back to Ahlfors that a real algebraic curve admits a real-fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, we are interested in characterising real algebraic varieties of dimension $n$ admitting real-fibered morphisms to the $n$-dimensional projective space. We present a criterion to classify real-fibered morphisms that arise as finite surjective linear projections from an embedded variety which relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real-fibered morphisms from real del Pezzo surfaces to the projective plane and determine which such morphisms arise as the composition of a projective embedding with a linear projection. Furthermore, we give some insights in the case of real conic bundles.