Recent advances in Brill--Noether theory and the geometry of Brill--Noether curves

TL;DR

This paper surveys recent progress in Brill--Noether theory, focusing on the moduli space of maps from curves to projective space and the geometry of Brill--Noether curves, while introducing key techniques developed recently.

Contribution

It provides a comprehensive overview of recent advances and introduces important techniques that have enabled these developments in Brill--Noether theory.

Findings

Progress in understanding the moduli space of maps from curves to projective space

New geometric insights into Brill--Noether curves

Development of innovative techniques in the last decade

Abstract

The first goal of this article is to survey recent progress in Brill--Noether theory, including both the study of the moduli space of maps from a curve to projective space and the geometry of the resulting curves in projective space. The second goal is to introduce newcomers to some of the important techniques that have been introduced or developed in the last decade that made these advances possible.