Narrow Results

We generalize a result of Flenner, proved in characteristic zero, to positive characteristics. We prove that the first syzygy bundle, , of the line bundle over is semistable, for a certain infinite set of integers d ≥ 0. Moreover, for arbitrary d, there is a "good enough estimate" on in terms of d and n; thus a strong restriction theorem of Langer, proved earlier for characteristic k > d, is valid in arbitrary characteristics.

We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

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 $H^0(L)\otimes {{𝒪}}_C\to L$. The exact sequence $0\to M_L\to H^0(L)\otimes {{𝒪}}_C\to L\to 0$ 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 $m_W:W^{\vee }\otimes H^0(K_C)\to H^0(S^{\vee }\otimes K_C)$, where $W\subseteq H^0(L)$ is a subspace and $S^{\vee }$ is the dual of a subbundle $S\subset M_L$. 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.