Pencils on real curves

TL;DR

This paper explores coverings of real algebraic curves to rational curves, establishing existence results with specific topological degrees, and investigates Brill-Noether theory for pencils on real curves, including coverings with degree 4 and prescribed covering numbers.

Contribution

It introduces new existence results for coverings with prescribed topological degrees and covering numbers, advancing Brill-Noether theory for real algebraic curves.

Findings

Existence of coverings with prescribed topological degree on real algebraic curves.

Construction of degree 4 coverings with specified covering number.

New insights into Brill-Noether theory for pencils on real curves.

Abstract

We consider coverings of real algebraic curves to real rational algebraic curves. We show the existence of such coverings having prescribed topological degree on the real locus. From those existence results we prove some results on Brill-Noether Theory for pencils on real curves. For coverings having topological degree 0 we introduce the covering number k and we prove the existence of coverings of degree 4 with prescribed covering number.