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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Here we prove the existence for a huge number of triples dY gY n of smooth complex projective curves X & P n with degree d, genus g and such that TP n jX is a stable vector bundle (approximatively, if g % n 2dan for n^4 and g % 2a27 1a2 d 3a2 for n 3).Let X & P n be a complex projective curve. Here we will study the stability and semistability of the restricted tangent bundle TP n jX. This topic was studied by several authors for certain integers nY d X deg XY g X p a X. For the semistability see in particular [14] and [15] for g 0, [10] for n 3, d^g 3, and references quoted there. One expects that the restricted tangent bundle is stable (or at least semistable) only if X is ªgeneralº in its component of the Hilbert scheme Hilb (P n ) of P n . However, there are further geometric obstructions and sometimes stronger stability results may be obtained. In this note we will show that the restricted tangent bundle is stable for a huge set of triples dY gY n. In particular (see Remark 1.9) for almost all dY gY n with g^2 we will show that the general non special embedding in P n of a general genus g curve has stable restricted tangent bundle. But our results are much sharper (see Theorem 2.12 for n 3 and Theorem 1.7 and Remark 1.8 for n b 3). If we look only at smooth curves with h 1 XY N X 0 our results are very strong and show a quite complete picture (see Remark 1.11 for the case n b 3, Remarks 2.14 and 2.15 for the case n 3). Our main tool is a result proved in [10], i.e. [10], Th. 2.4. Let A be a complex projective variety, E a vector bundle on A and C & A a reduced curve. In [10] it was defined the notion of semistability and stability for EjC. In [10], Th. 2.4, it was shown that the stability and the semistability of the restriction of E to flat families of reduced curves are open conditions. This was the main tool in [10] and [15]. Here (see Remark 1.2) we will show how to use it in a very efficient way.