Families of Elliptic Curves in P3 and Bridgeland Stability

Abstract

We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in the 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 blowup 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.