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 and irreducible 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(X ′ 10 ) of conics on a smooth ordinary Gushel-Mukai threefold with an irreducible component of a moduli space of Bridgeland stable objects in the Kuznetsov component of X ′ 10 . As a result, we show that the Kuznetsov's Fano threefold conjecture is not true.

Kuznetsov's Fano threefold conjecture via K3 categories and enhanced group actions

“…Remark 1.5. This result was also recently shown by Zhang [Zha21], via a completely different method, using uniqueness of (Serre-invariant) Bridgeland stability conditions and moduli spaces of stable objects.…”

“…There are alternative approaches to some of our results: Theorem 1.4 was proved independently by Zhang in [Zha21], based on a study of Bridgeland moduli spaces, while Theorem 1.6 was proved independently by Kuznetsov and Shinder in work in preparation [KS22] (see also [Kuz21,§5.4]), based on a degeneration argument and a theory of "absorption of singularities". Our paper and these two use completely different methods, which we believe are interesting in their own right.…”

Moduli spaces on the Kuznetsov component of Fano threefolds of index 2

“…Interestingly, this conjecture is incompatible with Kuznetsov's conjecture on Fano threefolds (see [Kuz09,Conjecture 3.7] for the conjecture, and [BT16, Theorem 4.2] for a proof of incompatibility). The conjecture on Fano threefolds has been recently disproven in [Zha20a].…”

Derived categories of hearts on Kuznetsov components

“…In the case of the quartic double solid it was observed in [BT16, Theorem 7.2] that Theorem 1.3 implies the failure of original Fano threefolds Kuznetsov's Conjecture [Kuz09, Conjecture 3.7]. Note that Fano threefolds Conjecture has been disproved in [Zha21] and [BP22], independently, in a stronger sense, namely that the Kuznetsov component of a quartic double solid is never equivalent to that of a Gushel-Mukai threefold.…”