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)-classes and (−4)-classes in the Kuznetsov components on index one prime Fano threefold X 4d+2 of degree 4d + 2 and index two prime Fano threefold Y d of degree d for 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)-classes on X 4d+2 and Y d are isomorphic. We show that moduli spaces of stable objects of (−1)-classes on X 14 are realized by Fano surface C(X) of conics, moduli spaces of semistable sheaves M X (2, 1, 6) and M X (2, −1, 6) and the correspondent moduli spaces on cubic threefold Y 3 are realized by moduli spaces of stable vector bundles M b Y (2, 1, 2) and M b Y (2, −1, 2). We show that moduli spaces of semistable objects of (−4)-classes on Y d are isomorphic to the moduli spaces of instanton sheaves M inst Y when d = 1, 2, and show that there're open immersions of M inst Y into moduli spaces of semistable objects of (−4)-classes when d = 1, 2. Finally, when d = 4, 5 we show that these moduli spaces are all isomorphic to M ss X (2, 0, 4).

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

“…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 ≥ 2 on a cubic threefold is irreducible.…”

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

“…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 [JLLZ21,LZ21] for many recent applications. In [FP21] the notion of Serre-invariance is applied to show that the moduli space of stable Ulrich bundles of rank d ě 2 on a cubic threefold is irreducible.…”

Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

“…Related works and motivations. The interest in the study of Serre-invariant stability conditions on semiorthogonal components in the bounded derived category has grown recently, mostly due to the applications to the study of moduli spaces (see for instance [PY20], [LZ21]) and to understand more 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 [BLMS17].…”

“…Another open question is to generalise Theorem 1.3 and Corollary 6.5 to further study of moduli spaces of semistable objects in Ku(X) with respect to a Serre-invariant stability condition like their projectivity or irreducibility. Some cases of small dimension have been studied in [PY20], [APR19], [BBF + 20], [Qin21], [LZ21].…”

“…The existence of Ulrich bundles have been shown on cubic threefolds by [LMS15, Theorem B]. In the rank-two case, namely the case of instanton sheaves of minimal charge, we know a full description of the moduli space with class 2[I ℓ ] for cubic threefolds [LMS15], and more generally for Fano threefolds of Picard rank 1, index 2, degree d ≥ 3 in [Qin21], [LZ21]. For the case of cubic fourfolds we refer to [FK20].…”