We show that the moduli space M X (v) of Gieseker stable sheaves on a smooth cubic threefold X with Chern character v = (3, −H, −H 2 /2, H 3 /6) is smooth and of dimension four. Moreover, the Abel-Jacobi map to the intermediate Jacobian of X maps it birationally onto the theta divisor Θ, contracting only a copy of X ⊂ M X (v) to the singular point 0 ∈ Θ.We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that X can be recovered from its Kuznetsov component Ku(X) ⊂ D b (X). Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, i.e., that X can be recovered from its intermediate Jacobian.
We study base points of the generalized Θ-divisor on the moduli space of vector bundles on a smooth algebraic curve X of genus g defined over an algebraically closed field. To do so, we use the derived categories D b (Pic 0 (X)) and D b (Jac(X)) and the equivalence between them given by the Fourier-Mukai transform FM P coming from the Poincaré bundle. The vector bundles P m on the curve X defined by Raynaud play a central role in this description. Indeed, we show that E is a base point of the generalized Θ-divisor, if and only if there exists a nontrivial homomorphism P rk(E)g+1 → E.
We construct vector bundles R rk µ on a smooth projective curve X having the property that for all sheaves E of slope µ and rank rk on X we have an equivalence: E is a semistable vector bundle ⇐⇒ Hom(R rk µ , E) = 0. As a byproduct of our construction we obtain effective bounds on r such that the linear system |R · Θ| has base points on U X (r, r(g − 1)).⊗rR(g−1)−dR 1 . This implies that F itself is a semistable bundle of slope 2.1 Construction of S µ,R,m for µ ∈ [−g − 1, −g)
We describe the birational correspondences, induced by the Fourier-Mukai functor, between moduli spaces of semistable sheaves on elliptic surfaces with sections, using the notion of P -stability in the derived category. We give explicit conditions to determine whether these correspondences are isomorphisms. This is indeed not true in general and we describe the cases where the birational maps are Mukai flops. Moreover, this construction provides examples of new compactifications of the moduli spaces of vector bundles via sheaves with torsion and via complexes. We finally get for any fixed dimension an isomorphism between the Picard groups of the moduli spaces.
We describe the possible restrictions of the cotangent bundle Ω P N to an elliptic curve C ⊂ P N . We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R +primary ideal I ⊂ R in the graded section ring R = n∈N Γ(C, O(n)).
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