Bridgeland Moduli spaces for Gushel-Mukai threefolds and Kuznetsov's Fano threefold conjecture
Abstract
We study the Hilbert scheme H of twisted cubics on a special smooth Gushel-Mukai threefolds X10. We show that it is a smooth irreducible projective threefold if X10 is general among special Gushel-Mukai threefolds, while it is singular if X10 is not general. We construct an irreducible component of a moduli space of Bridgeland stable objects in the Kuznetsov component of X10 as a divisorial contraction of H. We also identify the minimal model of Fano surface C(X10') of conics on a smooth ordinary Gushel-Mukai threefold with moduli space of Bridgeland stable objects in the Kuznetsov component of X10'. As a result, we show that the Kuznetsov's Fano threefold conjecture is not true
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