Abstract. A decorated vector bundle on a smooth projective curve X is a pair (E, ϕ) consisting of a vector bundle and a morphism ϕ : (E ⊗a ) ⊕b → (det E) ⊗c ⊗ N, where N ∈ Pic(X). There is a suitable semistability condition for such objects which has to be checked for any weighted filtration of E. We prove, at least when a = 2, that it is enough to consider filtrations of length ≤ 2. In this case decorated bundles are very close to quadric bundles and to check semistability condition one can just consider the former. A similar result for L-twisted bundles and quadric bundles was already proved ([1], [6]). Our proof provides an explicit algorithm which requires a destabilizing filtration and ensures a destabilizing subfiltration of length at most two. Quadric bundles can be thought as a generalization of orthogonal bundles. We show that the simplified semistability condition for decorated bundles coincides with the usual semistability condition for orthogonal bundles. Finally we note that our proof can be easily generalized to decorated vector bundles on nodal curves.

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Abstract. This paper is devoted to generalizing the Mehta-Ramanathan restriction theorem to the case of ε-semistable and k-semistable decorated sheaves. After recalling the definition of decorated sheaves and their usual semistability we define the ε and k-(semi)stablility. We first prove the existence of a (unique) ε-maximal destabilizing subsheaf for decorated sheaves (Section 3.1). After some others preliminar results (such as the opennes condition for families of ε-semistable decorated sheaves) we finally prove, in Section 3.7, a restriction theorem for slope ε-semistable decorated sheaves. In Section 4 we reach the same results in the k-semistability case that we did in the ε-semistability, but only for rank ≤ 3 decorated sheaves.

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In this article we study a special class of vector bundles, called tensors. A tensor consists of a vector bundle E over a smooth irreducible projective variety and a morphism of vector bundles ϕ. As for classical vector bundles, there exists a notion of stability for these objects given in terms of filtrations of the vector bundle E. The aim of the present paper is to prove that if a destabilizing filtration is "too" long then there exists a shorter subfiltration which destabilizes as well. Moreover, we describe some related combinatorial problems, which arise from the description of a tensor (E, ϕ) or, more precisely, a filtration of E as a a-dimensional matrix. Eventually, as example we study semistable tensors on the projective line.