# Maximal rank of space curves in the range A

## TL;DR

This paper proves a longstanding conjecture that for certain degrees and genera, there exist space curves in projective 3-space with maximal rank, confirming a key aspect of algebraic geometry related to Hilbert schemes.

## Contribution

It establishes the existence of irreducible components of the Hilbert scheme containing smooth space curves of maximal rank within specified degree and genus bounds.

## Key findings

- Existence of space curves with maximal rank for large degrees and genera.
- Confirmation of the conjecture from 1985.
- Identification of conditions for the irreducible components in the Hilbert scheme.

## Abstract

We prove the following statement, which has been conjectured since 1985: There exists a constant $K$ such that for all natural numbers $d,g$ with $g\le Kd^{3/2}$ there exists an irreducible component of the Hilbert scheme of $\mathbb{P}^3$ whose general element is a smooth, connected curve of degree $d$ and genus $g$ of maximal rank.