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

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.