Green's conjecture for general covers
Green's conjecture deals with the syzygies of a canonical curve \(\phi_{K_C} : C \rightarrow \mathbb P^{g-1}\). Specifically, it asserts the vanishing of certain Koszul cohomology groups: \(K_{p,2}(C,K_C) = 0 \Leftrightarrow p < \text{Cliff}(C)\). This implies that one can read off the Clifford index of the curve from the Betti diagram of the canonical embedding. Voisin established Green's conjecture for general curves \([C] \in \mathcal M_g\) of any genus. The first author found a sufficient condition for Green's conjecture to hold in terms of Brill-Noether theory: if \([C] \in \mathcal M_g\) is a \(d\)-gonal curve with \(2 \leq d \leq \frac{g}{2}+1\), and if \(\dim W_{g-d+2}^1(C) = \rho(g,1,g-d+2) = g-2d+2\), then \(C\) satisfies Green's conjecture. One aim of the current paper is to establish Green's conjecture in some classes of curves where the latter condition does not hold. This happens, for instance, for curves with an infinite number of minimal pencils. The second aim of the paper is to establish Green's conjecture for a broad class of curves with a fixed-point-free involution.NEWLINENEWLINEFor the entire collection see [Zbl 1236.14001].
scientific article; zbMATH DE number 6052333