# Deforming a canonical curve inside a quadric

## TL;DR

This paper investigates the deformation theory of canonical curves inside quadrics, establishing conditions for smoothness and obstructions, with implications for understanding the local structure of the Hilbert scheme.

## Contribution

It provides a detailed analysis of the deformation space of canonical curves within quadrics, including the dimension, smoothness criteria, and obstruction spaces, which was previously not fully understood.

## Key findings

- Hilbert scheme of the curve inside the quadric is a local complete intersection of dimension g^2-1
- The scheme is smooth when the quadric is smooth
- Obstructions are governed by the full Ext^1 group, especially when the curve meets the singular locus of the quadric

## Abstract

Let $C\subset{\mathbb P}^{g-1}$ be a canonically embedded nonsingular nonhyperelliptic curve of genus $g$ and let $X\subset{\mathbb P}^{g-1}$ be a quadric containing $C$. Our main result states among other things that the Hilbert scheme of $X$ is at $[C\subset X]$ a local complete intersection of dimension $g^2-1$, and is smooth when $X$ is. It also includes the assertion that the minimal obstruction space for this deformation problem is in fact the full associated $\operatorname{Ext}^1$-group and that in particular the deformations of $C$ in $X$ are obstructed in case $C$ meets the singular locus of $X$. As we will show in a forthcoming paper, this has applications of a topological nature.