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 ${\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}^{2m+2}({\mathbb{P}}^3)$ 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.