The locus in the moduli space of curves where the Petri map fails to be injective is called the Petri locus. In this paper we provide a new proof on the existence of Divisorial components in the Petri locus for the case of pencils. For this proof we produce some special reducible curves (chains of elliptic components) in the Petri locus and we show that such curves have only a finite number of pencils for which the Petri map is not injective.
Let C be a smooth irreducible complex projective curve of genus g and let B k (2, K C ) be the Brill-Noether locus parametrizing classes of (semi)stable vector bundles E of rank two with canonical determinant over C with h 0 (C, E) ≥ k. We show that B 4 (2, K C ) has an irreducible component B of dimension 3g −13 on a general curve C of genus g ≥ 8. Moreover, we show that for the general element [E] of B, E fits into an exact sequence 0 → O C (D) → E → K C (−D) → 0 with D a general effective divisor of degree three, and the corresponding coboundary map ∂ : H 0 (C, K C (−D)) → H 1 (C, O C (D)) has cokernel of dimension three.
Abstract. In the moduli space M g of smooth and complex irreducible projective curves of genus g, let GP g be the locus of curves that do not satisfy the GiesekerPetri theorem. Let GP 1 g,d be the subvariety of GP g formed by curves C of genus g with a pencil g
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
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