The Hilbert scheme of space curves with Hilbert polynomial 2t+2 has two components, parametrizing pairs of skew lines on the one hand, and conics union a point on the other. B. Schmidt has developed a technique for analysing Bridgeland wall crossings on threefolds, and applied this to twisted cubics and their deformations. Our example with skew lines and their deformations has similar behaviour, and Schmidt’s machinery may be applied and yields two wall crossings corresponding to a contraction of each of the two Hilbert scheme components. In this talk I seek to understand the two wall crossing conractions in terms of Mori theory, and find that each is a contraction of a K-negative extremal ray. In particular, all moduli spaces appearing are projective varieties.
Joint work with Sammy Soulimani.