green's canonical syzygy conjecture for generic curves of odd genus

By Claire Voisin. Compositio Mathematica, 2003

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we prove in this paper the green conjecture for generic curves of odd genus. that is, we prove the vanishing $k_{k,1}(x,k_x)=0$ for x a generic curve of genus $2k+1$. this completes our previous work, where the green conjecture for generic curves of genus g with fixed gonality d was proved in the range $d\geq g/3$, with the possible exception of the generic curves of odd genus. the case of generic curves of odd genus was considered as especially important, since hirschowitz and ramanan proved that if the conjecture is true for the generic curve of odd genus, then the locus of jumping syzygies is exactly the locus of exceptional gonality, as predicted by green's conjecture. thus, our result combined with the hirschowitz–ramanan result is a strong confirmation of green's conjecture.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6f737c49fb6591c70a57e1f99ce86ded5a01a5ae

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Citations in the corpus

Effective divisors on \bar{M}_g, curves on K3 surfaces and the Slope Conjecture (2003, methodology): “(cf. [Laz]). More recently, Voisin used the rich geometry of curves sitting on a K3 surface to prove Green’s Conjecture for general K3 sections and hence for the general curve of any genus (cf. [V2], [V3]). From the point of view of moduli spaces, the relevance of K in the study of the birational ge”
Koszul modules and Green’s conjecture (2018, methodology): “ff(C). Although for arbitrary curves the conjecture remains wide open, Green’s Conjecture for a general curve of every genus has been resolved using geometric methods in two landmark papers by Voisin [29, 30]. In this case one has Cliff(C) = ⌊g−1 2 ⌋and Green’s Conjecture reduces to a single vanishi”
The gonality conjecture on syzygies of algebraic curves of large degree (2014, unknown): “The purpose of this note is to show that a small variant of the methods used by Voisin in [13] and [14] leads to a surprisingly quick proof of the gonality conjecture of [9], asserting that one can read off the gonality of an algebraic curve C from its syzygies in the embedding defined by any one li”
Green’s conjecture for curves on arbitrary K3 surfaces (2009, methodology): “|L| satisﬁes the linear growthcondition (2), thus Green’s Conjecture is veriﬁed for any smooth curve in |L|. In the case ρ(g,1,k) = 1, Green’s Conjecture is also veriﬁed for smooth curves in |L|, cf. [V05], [HR98]. To sum up, Green’s Conjecture is veriﬁed for any curve of Clifford dimension one on a”