A note on the Kuznetsov component of the Veronese double cone

Abstract

This note describes moduli spaces of objects in the Kuznetsov component of a Veronese double cone, and some related constructions. We consider classes in the numerical Grothendieck group which are minimal with respect to the Euler form, and show that the corresponding moduli spaces are isomorphic. They are a blow-down of a moduli of tilt-stable objects. We interpret the latter space as a generalized Hilbert scheme of lines and give a wall-crossing interpretation of the blow-down morphism.

Categorical Torelli theorems for Gushel--Mukai threefolds

“…When F is not σ α,0 -semistable, the argument is slightly more complicated. Our main tools are inequalities in [PR20], [PY20, Proposition 4.1], and [Li16, Proposition 3.2], which allow us to bound the rank and first two Chern characters ch 1 , ch 2 of the destabilizing objects and their cohomology objects. Since F ∈ A X , by using the Euler characteristics χ(O X [1], −) and χ(−, O X (−1)[2]) we can obtain a bound on ch 3 .…”

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

“…More recently, the LLVS eightfold constructed in [LLVS17] and Fano variety of lines of cubic fourfold are constructed as moduli space of Bridgeland stable objects in Kuznetsov component on a cubic fourfold in [LPZ18] and [LMS18]. After a direct construction of stability condition in Kuznetsov components for a series Fano threefolds in [BLMS17], the moduli spaces of stable objects in the Kuznetsov components of prime Fano threefolds of index 2 are systematically studied in [APR19], [PY20], [PR20] and [BBF+20]. In [PY20], the author show that Σ(Y d ) is constructed as moduli space of Bridgeland stable objects in Ku(Y d ) for any d = 1 while Σ(Y 1 ) is an irreducible component of moduli space of stable objects in Ku(Y 1 ) [PR20].…”

“…After a direct construction of stability condition in Kuznetsov components for a series Fano threefolds in [BLMS17], the moduli spaces of stable objects in the Kuznetsov components of prime Fano threefolds of index 2 are systematically studied in [APR19], [PY20], [PR20] and [BBF+20]. In [PY20], the author show that Σ(Y d ) is constructed as moduli space of Bridgeland stable objects in Ku(Y d ) for any d = 1 while Σ(Y 1 ) is an irreducible component of moduli space of stable objects in Ku(Y 1 ) [PR20]. In our paper, we show that Σ(X 10 ) can be realized as a subvariety of Z and minimal model of Fano surface C m (X ′ 10 ) of conics can be realized as an irreducible component of moduli space of stable objects in A X ′ 10 .…”

Categorical Torelli theorems: results and open problems

“…This makes this case quite different from the others. A remarkable difference is that the Fano surfaces of lines is an irreducible component of a moduli space of stable objects in Ku(Y 1 ) and in [116] the authors classify all the objects in this moduli space. We can then formulate the following questions: We now focus on the index 1 case and denote by X d such a prime Fano threefold of degree d = 2g − 2, where 2 ≤ g ≤ 12, g = 11.…”