For general quartic double solids, two varieties are isomorphic if and only if their Kuznetsov components are equivalent, without assuming the equivalence has Fourier-Mumford type.
Stability conditions, Bogomolov-Gieseker type inequalities and Fano 3-folds
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abstract
We develop a framework to modify the Bogomolov-Gieseker type inequality conjecture introduced by Bayer-Macri-Toda, in order to construct a family of geometric Bridgeland stability conditions on any smooth projective 3-fold. We show that it is enough to check these modified inequalities on a small class of tilt stable objects. We extend some of the techniques in the works by Li and Bernardara-Macri-Schmidt-Zhao to formulate a strong form of Bogomolov-Gieseker inequality for tilt stable objects on Fano 3-folds. Consequently, we establish our modified Bogomolov-Gieseker type inequality conjecture for general Fano 3-folds, including an optimal inequality for the blow-up of $P^3$ at a point.
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2019 1verdicts
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Moduli spaces on the Kuznetsov component of Fano threefolds of index 2
For general quartic double solids, two varieties are isomorphic if and only if their Kuznetsov components are equivalent, without assuming the equivalence has Fourier-Mumford type.