Bridgeland Moduli spaces for Gushel-Mukai threefolds and Kuznetsov's Fano threefold conjecture

Abstract

We study the Hilbert scheme $\mathcal{H}$ of twisted cubics on a special smooth Gushel-Mukai threefolds $X_{10}$. We show that it is a smooth irreducible projective threefold if $X_{10}$ is general among special Gushel-Mukai threefolds, while it is singular if $X_{10}$ is not general. We construct an irreducible component of a moduli space of Bridgeland stable objects in the Kuznetsov component of $X_{10}$ as a divisorial contraction of $\mathcal{H}$. We also identify the minimal model of Fano surface $\mathcal{C}(X_{10}^{\prime })$ of conics on a smooth ordinary Gushel-Mukai threefold with moduli space of Bridgeland stable objects in the Kuznetsov component of $X_{10}^{\prime }$. As a result, we show that the Kuznetsov's Fano threefold conjecture is not true

35 pages, substantially rewritten, remove everything irrelevant. Fix typos and correct mistakes on computations, prove stronger results via Serre-invariant stability conditions

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Cited by in corpus (2)

• Serre-invariant stability conditions and Ulrich bundles on cubic threefolds
• Serre functors and dimensions of residual categories