Some geometric properties of Hilbert schemes of space curves

TL;DR

This paper provides an explicit description of the geometric cones of the Hilbert scheme of space curves in complex projective 3-space, and explores implications for its automorphism group.

Contribution

It offers a complete, explicit characterization of the cone of curves and the ample cone of the Hilbert scheme of space curves, advancing understanding of its geometry.

Findings

Explicit description of the cone of curves

Explicit description of the ample cone

Partial results on the automorphism group

Abstract

Let $H$ be the Hilbert scheme of curves in complex projective $3$-space, with $d\geq 3$ and genus $g \leq (d-2)^2/4$. A complete, explicit description of the cone of curves and the ample cone of $H$ is given. From this, partial results on the group $\mathop{Aut}(H)$ are deduced.