Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface

By Claire Voisin. 2002

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for X a variety and L a line bundle on X. Denoting by Kp,q(X,L) the cohomology at the middle of the sequence above, one sees immediately that the surjectivity of the map μ0 is equivalent to K0,2(C, KC) = 0, and that if this is the case, the ideal I is generated by quadrics if and only if K1,2(C, KC) = 0. On the other hand, C being non hyperelliptic is equivalent to the fact that the Clifford index Cliff C is strictly positive, where

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2cdd3db33cce43c7a8075b21fd62d8d5aefa2752

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Citations in the corpus

Moduli of theta-characteristics via Nikulin surfaces (2011, background): “Fundamental results in the theory of algebraic curves like the Brill-Noether Theorem [21], or Green’s Conjecture for generic curves [30] have been proved by specialization to a general point [C] ∈ Kg .”
green's canonical syzygy conjecture for generic curves of odd genus (2003, background): “The following key point, which was used constantly throughout the proof of [12], remains true in our situation : Proposition 2 The determinant map”
Lectures on K-theoretic computations in enumerative geometry (2015, background): “See for example [24,85] for places in the literature where a much more powerful calculus of this kind is explained and used.”
Koszul divisors on moduli spaces of curves (2006, background): “Although Green’s Conjecture for arbitrary curves is still open, Voisin proved it for generic curves of given gonality (cf. [ V1 ], [V2]). In our case this gives 822”
Pfaffian quartic surfaces and representations of Clifford algebras (2011, methodology): “been used extensively to study the interplay between curves and vector bundles on K3 surfaces, starting with [Laz]. It plays a major role in Voisin’s proof of the Green conjecture for generic curves ([Voi1],[Voi2]), and in the recent proof of the Green conjecture for curves on arbitrary K3 surfaces ”
Green’s conjecture for curves on arbitrary K3 surfaces (2009, methodology): “lowing: Theorem 1.1. Let S be a K3 surface and C ⊂ S be a smooth curve with g(C) = g and gon(C) = k. If k ≤ (g +2)/2, then C satisﬁes Green’s conjecture. Note that Theorem 1.1 has been established in [V02] when Pic(S) = Z· C. The proof relies (via [Ap05]) on the case of curves of odd genus of maxima”
The gonality conjecture on syzygies of algebraic curves of large degree (2014, unknown): “1 Using Voisin’s results [13, 14] on syzygies of general canonical curves, Aprodu and Voisin [1, 3] proved the statement of the theorem for a general curve of each gonality.”
The Kodaira dimension of the moduli space of Prym varieties (2008, background): “(see also [GL], [L], [ Vo ], [F1], [FMP] for many applications of these bundles).”
Koszul modules and Green’s conjecture (2018, methodology): “eorem 1.3 using a suitable positive characteristic version of Bott vanishing, in conjunction with the hypercohomology spectral sequence that was featured in the work of Voisin on the Green conjecture [29]. We recall the basics on partitions and Schur functors in Section 2.1, and use them to formulat”
On integral Hodge classes on uniruled or Calabi-Yau threefolds (2004, unknown): “which are partially computed in [15].”
Computational Algebraic Geometry (2003, background): “For substantial recent progress on this, see Teixidor [94] and Voisin [96].”