On a Conjecture of Butler
Syzygy bundles over smooth curves (as well as higher dimensional smooth varieties) have been studied for several years now. Let L be a line bundle on a smooth curve X. Given a subspace V of the space of sections of L which generates L, the kernel M L,V of
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Introduction Syzygy bundles over smooth curves (as well as higher dimensional smooth varieties) have been studied for several years now. Let L be a line bundle on a smooth curve X. Given a subspace V of the space of sections of L which generates L, the kernel ML,V of the evaluation map V ⊗ OX → L is called a Syzygy bundle or a Kernel bundle or a Lazarfeld bundle. These bundles have several applications, applications to Syzygy problems, Greens conjectures, Minimal Resolution conjectures, Theta functions, Picard bundles. They also play an important role in Brill-Noether theory for higher ranks and coherent systems. Eighteen years back, D.C. Butler made a conjecture about the semistability of ML,V for general (L, V ) [15]. The conjecture was proved recently by Peter Newstead, myself and Leticia Brambila-Paz [8]. In this article, we first present a short survey of previous work on the conjecture of Butler and related conjectures. Then we state our main results and sketch the idea of the proofs. We end with a discussion of some applications of Kernel bundles and generalisations of some results on Kernel bundles to nodal and cuspidal curves.
2. Preliminaries Let X be an integral smooth projective curve of genus g. Let E denote a vector bundle of rank n on X and p : E → X the projection map. For x ∈ X, the fibre Ex := p−1 x is a vector space of dimension n. We recall that a section s ∗ Department of Mathematics, Indian Institute of Science, Bangalore 560012, India. E-mail: [email protected] † This article was written during author’s tenure as Raja Ramanna Fellow in Indian Institute of Science, Bangalore. It was finalised during the visit of the author to Instituto de Ciencias Matematicas under the Indo-European Mary Curie research program MODULI in June-July 2015. The author thanks both the institutes for excellent working conditions.
© Springer Nature Singapore Pte Ltd. 2017 and Hindustan Book Agency 2017 A. Aryasomayajula et al. (eds.), Analytic and Algebraic Geometry, DOI 10.1007/978-981-10-5648-2_3
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Usha N. Bhosle
of E is a map X → E such that p ◦ s = idX . Let H 0 (X, E) be the space of sections of E, it is a finite dimensional vector space.
2.1. (semi)stability of vector bundles. Associated to any vector bundle E, there is an integer d(E) called the degree of E. Denote by r(E) the rank of E (r(E) = n). Define slope of E by μ(E) := d(E)/r(E) . A sub bundle F of E is a vector bundle F such that F ⊂ E with each fibre Fx ⊂ Ex for all x ∈ X. A sub bundle F is called proper if F = E. Definition 2.1. A vector bundle E is called semistable if for every sub bundle F ⊂ E, μ(F ) ≤ μ(E) . A vector bundle F is called stable if for every proper sub bundle F of E, μ(F ) < μ(E).
2.2. Syzygy Bundle. Let V ⊂ H 0 (X, E) be a vector subspace and X ×V denote the trivial bundle with fibre isomorphic to V . Definition 2.2. Define the evaluation map evV : X × V → E by (x, s) → s(x), x ∈ X, s ∈ V. If this map is surjective, then E is said to be generated by V . Definition 2.3. Let L denote a line bundle of degree d on X. Suppose that the
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