# Differential geometry of Hilbert schemes of curves in a projective space

## TL;DR

This paper explores the geometric structure of Hilbert schemes of curves within projective spaces, providing insights into their natural geometric properties in three and higher dimensions.

## Contribution

It introduces a detailed description of the natural geometry of Hilbert schemes of curves in projective spaces, extending understanding to higher dimensions.

## Key findings

- Characterization of the geometry of Hilbert schemes in ${f P}^3$
- Extension of geometric descriptions to ${f P}^n$, $n extgreater 3$
- Insights into the structure of curve families in projective spaces

## Abstract

We describe the natural geometry of Hilbert schemes of curves in ${\mathbb P}^3$ and, in some cases, in ${\mathbb P}^n$ , $n\geq 4$.