Bridgeland stability conditions and skew lines on $\mathbb{P}^3$

TL;DR

This paper explores wall crossings in Bridgeland stability conditions on $P^3$, analyzing moduli spaces of pairs of skew lines and conics, revealing two walls and their geometric implications.

Contribution

It identifies and describes two specific wall crossings in Bridgeland stability for certain moduli spaces, showing their contractions are smooth and $K$-negative extremal.

Findings

Two walls in Bridgeland stability were found.

Each wall crossing results in a smooth divisor contraction.

The contractions are $K$-negative extremal, ensuring projectivity.

Abstract

Inspired by Schmidt's work on twisted cubics, we study wall crossings in Bridgeland stability, starting with the Hilbert scheme $\mathrm{Hilb}^{2m+2}(\mathbb{P}^3)$ parametrizing pairs of skew lines and plane conics union a point. We find two walls. Each wall crossing corresponds to a contraction of a divisor in the moduli space and the contracted space remains smooth. Building on work by Chen--Coskun--Nollet we moreover prove that the contractions are $K$-negative extremal in the sense of Mori theory and so the moduli spaces are projective.