Brill-Noether-type Theorems with a Movable Ramification Point

TL;DR

This paper extends classical Brill-Noether theorems to include ramification conditions at a movable point, providing complete solutions in low dimensions and bounds in general.

Contribution

It introduces a new approach to handle ramification at an unspecified point, expanding the classical theory to more flexible conditions.

Findings

Complete solutions for dimensions 1 and 2.

Existence test and dimension bounds in general case.

Extension of classical theorems to movable ramification points.

Abstract

The classical Brill-Noether theorems count the dimension of the family of maps from a general curve of genus g to non-degenerate curves of degree d in r-dimensional projective space. These theorems can be extended to include ramification conditions at fixed general points. This paper deals with the problem of imposing a ramification condition at an unspecified point. We solve the problem completely in dimensions 1 and 2, and provide an existence test and bound the dimension of the family in the general case.