A note on Bridgeland moduli spaces and moduli spaces of sheaves on $$X_{14}$$ and $$Y_3$$

Abstract

We study Bridgeland moduli spaces of semistable objects of $$(-1)$$ ( - 1 ) -classes and $$(-4)$$ ( - 4 ) -classes in the Kuznetsov components on index one prime Fano threefold $$X_{4d+2}$$ X 4 d + 2 of degree $$4d+2$$ 4 d + 2 and index two prime Fano threefold $$Y_d$$ Y d of degree d for $$d=3,4,5$$ d = 3 , 4 , 5 . For every Serre-invariant stability condition on the Kuznetsov components, we show that the moduli spaces of stable objects of $$(-1)$$ ( - 1 ) -classes on $$X_{4d+2}$$ X 4 d + 2 and $$Y_d$$ Y d are isomorphic. We show that moduli spaces of stable objects of $$(-1)$$ ( - 1 ) -classes on $$X_{14}$$ X 14 are realized by Fano surface $$\mathcal {C}(X)$$ C ( X ) of conics, moduli spaces of semistable sheaves $$M_X(2,1,6)$$ M X ( 2 , 1 , 6 ) and $$M_X(2,-1,6)$$ M X ( 2 , - 1 , 6 ) and the correspondent moduli spaces on cubic threefold $$Y_3$$ Y 3 are realized by moduli spaces of stable vector bundles $$M^b_Y(2,1,2)$$ M Y b ( 2 , 1 , 2 ) and $$M^b_Y(2,-1,2)$$ M Y b ( 2 , - 1 , 2 ) . We show that moduli spaces of semistable objects of $$(-4)$$ ( - 4 ) -classes on $$Y_{d}$$ Y d are isomorphic to the moduli spaces of instanton sheaves $$M^{inst}_Y$$ M Y inst when $$d\ne 1,2$$ d ≠ 1 , 2 , and show that there are open immersions of $$M^{inst}_Y$$ M Y inst into moduli spaces of semistable objects of $$(-4)$$ ( - 4 ) -classes when $$d=1,2$$ d = 1 , 2 . Finally, when $$d=3,4,5$$ d = 3 , 4 , 5 we show that these moduli spaces are all isomorphic to $$M^{ss}_X(2,0,4)$$ M X ss ( 2 , 0 , 4 ) .

Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

“…The interest in the study of Serre-invariant stability conditions on semiorthogonal components in the bounded derived category has grown recently, mostly due to applications to the study of moduli spaces (see for instance [PY20,LZ22]) and to the desire to better understand the Kuznetsov component. For Fano threefolds of Picard rank 1 and index 2, the existence of Serre-invariant stability conditions on their Kuznetsov component is proved in [PY20], making use of the stability conditions constructed in [BLM + 17].…”

“…The existence of Ulrich bundles have been shown on cubic threefolds by [LMS15, Theorem B]. In the rank 2 case, namely the case of instanton sheaves of minimal charge, we know a full description of the moduli space with class 2[I ] for cubic threefolds, see [LMS15], and more generally for Fano threefolds of Picard rank 1, index 2 and degree d ≥ 3; see [Qin21,LZ22]. For the case of cubic fourfolds, we refer to [FK20].…”

Categorical Torelli theorems for Gushel–Mukai threefolds

“…In [51], the authors realize the Fano surface of lines Σ(𝑌 𝑑 ) (for 𝑑 ⩾ 2) as a Bridgeland moduli space of stable objects in the Kuznetsov component 𝑢(𝑌 𝑑 ). In [40], the authors realize the moduli space of rank two instanton sheaves on a del Pezzo threefold 𝑌 𝑑 (for 𝑑 ⩾ 3) and the compactification of the moduli space of ACM sheaves on 𝑋 4𝑑+2 (for 𝑑 ⩾ 3) as Bridgeland moduli spaces of stable objects in 𝑢(𝑌 𝑑 ) and 𝑢(𝑋 4𝑑+2 ), respectively. In [17], the authors realize the moduli space of Ulrich bundles of arbitrary rank on a cubic threefold 𝑌 3 as an open locus of a Bridgeland moduli space of stable objects in 𝑢(𝑌 3 ).…”

“…Serre‐invariance is one of the fundamental tools in studying relationship of classical Gieseker moduli spaces and Bridgeland moduli spaces for Kuznetsov components (cf. [1, 17, 40, 51, 54]). A natural question is whether any two Serre‐invariant stability conditions are in the same $$\widetilde{\mathrm{GL}}^+(2,\mathbb {R})$$‐orbit.…”

Stability conditions on Kuznetsov components of Gushel–Mukai threefolds and Serre functor

“…They also gave another proof of the categorical Torelli Theorem for cubic threefolds in [41, Theorem 5.17], following the strategy in [8, Theorem 1.1], where this result was proved for the first time (see also [6] for a different approach). In fact, the property of Serre‐invariance is very helpful in the study of the properties of moduli spaces and the stability of objects, see, for instance, [18, 32] for many recent applications. In [14] the notion of Serre‐invariance is applied to show that the moduli space of stable Ulrich bundles of rank $$d \ge 2$$ on a cubic threefold is irreducible.…”