Let C be a smooth irreducible projective curve and let (L, H 0 (C, L)) be a complete and generated linear series on C. Denote by ML the kernel of the evaluation map H 0 (C, L) ⊗ OC → L. The exact sequence 0 → ML → H 0 (C, L) ⊗ OC → L → 0 fits into a commutative diagram that we call the Butler's diagram. This diagram induces in a natural way a multiplication map on global sections mW :is a subspace and S ∨ is the dual of a subbundle S ⊂ ML. When the subbundle S is a stable bundle, we show that the map mW is surjective. When C is a Brill-Noether general curve, we use the surjectivity of mW to give another proof on the semistability of ML, moreover we fill up a gap of an incomplete argument by Butler: With the surjectivity of mW we give conditions to determinate the stability of ML, and such conditions implies the well known stability conditions for ML stated precisely by Butler. Finally we obtain the equivalence between the stability of ML and the linear stability of (L, H 0 (L)) on γ-gonal curves.The bundle M V,L is called Lazarsfeld-Mukai bundle. When V = H 0 (C, L), we will denote the bundle M H 0 (L),L by M L . The vector bundle M V,L and its dual M ∨ V,L have been studied from different points of view because of the rich geometry they encode. The study of the stability of M V,L is related with: the study of Brill-Noether varieties (see [4]), the Resolution Minimal Conjecture (see [6]), the stability of the tangent bundle of a projective space restricted to a curve; and the theta divisors of vector bundles on curves (see [7], [8]). Ein and Lazarsfeld used the stability of M V,L to prove the stability of the Picard bundle (see [5]). In ([10]), Paranjape and Ramanan proved that M K C is semistable, and David C. Butler showed that M L is stable for d > 2g, and it is semistable for d = 2g (see [2] and [10]).David Mumford introduced the concept of linear stability for projective varieties X ⊂ P n (see [9]). In some sense, this definition is a way to measure how X sits in P n . It was generalized for linear series (L, V ) over a curve C (see [8]). Linear stability of a generated linear series (L, V ) is a weaker condition than the stability for the vector bundle M V,L , that is, the stability of M V,L

In the last decades there have been introduced different concepts of stability for projective varieties. In this paper we give a natural and intrinsic criterion of the Chow, and Hilbert, stability for complex irreducible smooth projective curves C ⊂ P n . Namely, if the restriction T P n |C of the tangent bundle of P n to C is stable then C ⊂ P n is Chow stable, and hence Hilbert stable. We apply this criterion to describe a smooth open set of the irreducible component Hilb P (t),s Ch of the Hilbert scheme of P n containing the generic smooth Chow-stable curve of genus g ≥ 4 and degree d > g + n − g n+1 . Moreover, we describe the quotient stack of such curves. Similar results are obtained for the locus of Hilbert stable curves.

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Let C be a smooth complex irreducible projective curve of genus g with general moduli, and let (L, H 0 (L)) be a generated complete linear series of type (d, r + 1) over C. The syzygy bundle, denoted by ML, is the kernel of the evaluation map H 0 (L) ⊗ OC → L. In this work we have a double purpose. The first one is to give new examples of stable syzygy bundles admitting theta divisor over general curves. We prove that if ML is strictly semistable then ML admits reducible theta divisor. The second purpose is to study the cohomological semistability of ML, and in this direction we show that when L induces a birational map, the syzygy bundle ML is cohomologically semistable, and we obtain precise conditions for the cohomological semistability of ML where such conditions agree with the semistability conditions for ML.

In previous works, the authors investigated the relationships between linear stability of a generated linear series |V| on a curve C, and slope stability of the syzygy vector bundle $$M_{V,L} := \ker (V \otimes \mathcal {O}_C \rightarrow L)$$ M V , L : = ker ( V ⊗ O C → L ) . In particular, the second named author and L. Stoppino conjecture that, for a complete linear system |L|, linear (semi)stability is equivalent to slope (semi)stability of $$M_{L}$$ M L . The first and third named authors proved that this conjecture holds in the two opposite cases: hyperelliptic and generic curves. In this work we provide a counterexample to this conjecture on any smooth plane curve of degree 7.