On Enriques-Babbage Theorem for singular curves

TL;DR

This paper extends the Enriques-Babbage Theorem to singular curves by analyzing their canonical models, providing partial proofs and explicit descriptions for specific classes of singular curves.

Contribution

It introduces a version of the Enriques-Babbage Theorem applicable to singular curves and offers a complete proof for unicuspidal monomial curves.

Findings

Partial proof of the theorem for arbitrary singular curves.

Complete proof for unicuspidal monomial curves.

Explicit description of the ideal generators for the canonical model.

Abstract

We propose a version of the Enriques-Babagge Theorem for a singular curve $C$, involving its canonical model $C'$. We provide a partial proof for an arbitrary curve $C$ and complete the proof for unicuspidal monomial curves by describing the generators of the ideal of $C'\subset\mathbb{P}^{g-1}$.