Universal secant bundles and syzygies of canonical curves
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Universal secant bundles and syzygies of canonical curves Michael Kemeny1
Received: 8 April 2020 / Accepted: 27 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We introduce a relativization of the secant sheaves from Green and Lazarsfeld (A simple proof of Petri’s theorem on canonical curves, Geometry Today, 1984) and Ein and Lazarsfeld (Inventiones Math 190:603-646, 2012) and apply this construction to the study of syzygies of canonical curves. As a first application, we give a simpler proof of Voisin’s Theorem for general canonical curves. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves. Secondly, in the case of curves of even genus, we enhance Voisin’s Theorem by providing a structure theorem for the last syzygy space, resolving the Geometric Syzygy Conjecture in even genus. 0 Introduction In this paper, we introduce a universal version of the secant sheaf construction from [13]. Using this tool, we give a simpler proof of a theorem of Voisin [28], [29] on the equations of canonical curves. We further give a generalization of her result for curves of even genus. The classical Theorem of Noether–Babbage–Petri states that canonical curves are projectively normal, and that the ideal IC/Pg−1 is generated by quadrics (with a few exceptions), see [6] for a modern treatment. In the 1980s,
B Michael Kemeny
[email protected]
1
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln Dr, Madison, WI 53706, USA
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M. Kemeny
M. Green realized that these classical results about the equations defining canonical curves should be the first case of a much more general statement about higher syzygies, and he made a very influential conjecture [17] in this direction. Whilst the general case of Green’s Conjecture remains open, in 2002 Voisin made a breakthrough by proving the conjecture for general curves of even genus [28]. Voisin’s argument relies on an intricate study of the geometry of Hilbert schemes on a K3 surface. Recently, an algebraic approach to Voisin’s Theorem has been given, [3], based on degenerating to the tangent developable, a singular surface whose hyperplane sections are cuspidal curves. The authors apply the representation theory of an S L 2 action present in this special situation to establish Green’s conjecture for rational cuspidal curves. Explicit plethysm formulae play the key role, involving a change of basis between elementary symmetric polynomials and Schur polynomials. Maps which are simple to describe in one basis become rather complicated in the other, making the proof quite technical, see [3, Sect. 5.5–5.7]. In this paper, we first give a simpler proof of Voisin’s Theorem, using only basic homological algebra and without the need to degenerate. We further provide a structure theorem in the even genus case, describing in detail the extremal syzygy space. Let X be a complex K3 surface with Picard group generated by an ample line bundle L of even genus g = 2k, i.e. (L)2 = 2g −2.
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