The Osculating cone to special Brill-Noether Loci

TL;DR

This paper investigates the osculating cone to Brill-Noether loci on a smooth curve, revealing that the canonical curve is a component of this cone, using classical Kempf techniques.

Contribution

It provides a detailed description of the osculating cone to Brill-Noether loci at specific points, highlighting the canonical curve as a component.

Findings

Canonical curve is a component of the osculating cone.

Description of the osculating cone at smooth isolated points.

Application of Kempf's techniques to this geometric setting.

Abstract

In this paper, we describe the osculating cone to Brill-Noether loci $W^{0}_{d}(C)$ at smooth isolated points of $W^{1}_{d}(C)$ for a smooth canonically embedded curve $C$ of even genus $g=2(d-1)$. In particular, we show that the canonical curve $C$ is a component of the osculating cone. The proof is based on techniques introduced by George Kempf in 1986.