Bridgeland stability conditions and skew lines on ℙ³

Abstract

Inspired by Schmidt’s work on twisted cubics, we study wall crossings in Bridgeland stability, starting with the Hilbert scheme ${\text{Hilb}}^{2m+2}({\mathbb{P}}^3)$ 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.

KEYWORDS:

• Bridgeland stability
• Hilbert scheme
• Mori cone
• skew lines
• wall crossing

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 14C05
• Secondary: 14D20
• 14E30
• 14F08

Additional information

Funding

This work is supported by the Research Council of Norway under Grant No. 230986, and is part of the Ph.D. thesis [1] defended by the first author at the university of Stavanger on the 29th of March 2023. The second author was also supported by the Swedish Research Council under Grant No. 2016-06596 while the author was in residence at Institut Mittag–Leffler in Djursholm, Sweden during the fall of 2021.