Segre embeddings and the canonical image of a curve

TL;DR

This paper proves that for a general algebraic curve of any genus, its canonical embedding cannot be realized within a Segre embedding of a product of three or more projective spaces, revealing limitations of such embeddings.

Contribution

It establishes a new non-existence result for canonical embeddings of general curves within certain Segre embeddings, extending understanding of curve embeddings.

Findings

No general curve's canonical image lies on a Segre embedding of three or more projective spaces.

The result applies to all genera g, indicating a fundamental geometric restriction.

Provides insight into the structure of canonical embeddings and their limitations.

Abstract

We prove that there is no g for which the canonical embedding of a general curve of genus g lies on the Segre embedding of any product of three or more projective spaces.