Families of elliptic curves in $\mathbb{P}^3$ and Bridgeland Stability

TL;DR

This paper investigates wall crossings in Bridgeland stability for elliptic quartic curves in projective space, describing moduli spaces and computing the effective cone, with new geometric insights and proofs.

Contribution

It provides a geometric description of moduli spaces of elliptic quartic curves and offers a novel proof of the principal component's structure as a double blow-up.

Findings

Described the moduli spaces via wall crossings.

Computed the cone of effective divisors.

Provided a new proof of the principal component's structure.

Abstract

We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in three dimensional projective space. We provide a geometric description of each of the moduli spaces we encounter, including when the second component of this Hilbert scheme appears. Along the way, we prove that the principal component of this Hilbert scheme is a double blow up with smooth centers of a Grassmannian, exhibiting a completely different proof of this known result by Avritzer and Vainsencher. This description allows us to compute the cone of effective divisors of this component.