Linear stability and stability of syzygy bundles.

Let C be a smooth irreducible projective curve and let (L , H 0 (L)) be a complete and generated linear series on C. Denote by M L the kernel of the evaluation map H 0 (L) ⊗ 𝒪 C → L. The exact sequence 0 → M L → H 0 (L) ⊗ 𝒪 C → 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 m W : W ∨ ⊗ H 0 (K C) → H 0 (S ∨ ⊗ K C) , where W ⊆ H 0 (L) is a subspace and S ∨ is the dual of a subbundle S ⊂ M L . When the subbundle S is a stable bundle, we show that the map m W is surjective. When C is a Brill–Noether general curve, we use the surjectivity of m W to give another proof of the semistability of M L , moreover, we fill up a gap in some incomplete argument by Butler: With the surjectivity of m W we give conditions to determine the stability of M L , and such conditions imply the well-known stability conditions for M L stated precisely by Butler. Finally we obtain the equivalence between the (semi)stability of M L and the linear (semi)stability of (L , H 0 (L)) on k -gonal curves. [ABSTRACT FROM AUTHOR]