arXiv subjects
Publications and source records attributed to Michael Kemeny.
In this note, we give a new proof of Voisin's theorem on Green's conjecture for generic curves of odd genus resembling the first two sections of "Universal Secant Bundles and Syzygies of Canonical Curves" by the author, and so avoiding the need for difficult computations.
We show the geometric syzygy conjecture in positive characteristic. Specifically, if C is a general smooth curve of genus g defined over an algebraically closed field of characteristic p, then all linear syzygy spaces are spanned by syzygies of minimal rank provided p is at least 2g-4.
We study the Ein-Lazarsfeld Conjecture of syzygies of Veronese varieties in the case of linear syzygies q=1. We show a vanishing statement which agrees with the conjecture up to highest and second-highest order for linear syzygies.
We prove that the linear syzygy spaces of a general canonical curve are spanned by syzygies of minimal rank.
We construct Eagon--Northcott cycles on Hurwitz space and compare their classes to Kleiman's multiple point loci. Applying this construction towards the classification of Betti tables of canonical curves, we find that the value of the extremal Betti number records the number of minimal pencils. The result holds under transversality hypotheses equivalent to the virtual cycles having a geometric interpretation. We analyse the case of two minimal pencils, showing that the transversality hypotheses hold generically.
We introduce a relativization of the secant sheaves used by Ein, Green and Lazarsfeld 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.
We explore the concept of projections of syzygies and prove two new technical results; we firstly give a precise characterization of syzygy schemes in terms of their projections, secondly, we prove a converse to Aprodu's Projection Theorem. Applying these results, we prove that extremal syzygies of general curves of non-maximal gonality embedded by a linear system of sufficiently high degree arise from scrolls. Lastly, we prove Green's Conjecture for general covers of elliptic curves (of arbitrary degree) as well as proving a new result for curves of even genus and maximal gonality.
We give a simple proof of Voisin's Theorem for general canonical curves of even genus. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves.
We prove two statements concerning the linear strand of the minimal free resolution of a curve of fixed gonality. Firstly, we show that a general curve C of genus g of non-maximal gonality k\leq (g+1)/2 satisfies Schreyer's Conjecture, that is, b_{g-k,1}(C,K_C)=g-k, and all its highest order linear syzygies are of Eagon-Northcott type. This is a statement going beyond Green's Conjecture and predicts that all highest order linear syzygies in the canonical embedding of C are determined by the syzygies of the (k-1)-dimensional scroll containing C. Secondly, we formulate an optimal effective version of the Gonality Conjecture and prove it for general k-gonal curves. This generalizes the asymptotic Gonality Conjecture proved by Ein-Lazarsfeld and improves results of Rathmann in the case where C is a general curve of fixed gonality.
We prove the Geometric Syzygy Conjecture for generic canonical curves of even genus. This result extends Green's classical result on the generation of the ideal of a canonical curve by rank four quadrics to the highest linear syzygy group.
The Prym-Green conjecture predicts that the resolution of a general level p paracanonical curve of genus g is natural. Using decomposable ruled surfaces over an elliptic curve, we provide a complete solution (that is, for all levels) to this conjecture in odd genus.
We survey three results on syzygies of curves beyond Green's conjecture, with a particular emphasis on drawing connections between the study of syzygies and other topics in moduli theory.
The Prym-Green Conjecture predicts that the resolution of a generic n-torsion paracanonical curve of every genus is natural. The conjecture has mostly been studied so far for level 2, that is, for Prym-canonical curves. Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove the Prym-Green Conjecture for curves of odd genus g and torsion bundles of sufficiently high order with respect to g. We also give partial results in even genus. In the process, we confirm the expectation of Barth and Verra concerning the number of curves in a fixed linear system on a K3 surface, having an n-torsion line bundle induced by restriction from the K3 surface.
Let $C$ be a curve and $L$ a very ample line bundle. The Green-Lazarsfeld Secant conjecture predicts that if the degree of $L$ is at least $2g+p+1-2h^1(C,L)-Cliff(C)$ and if, in addition, $L$ is $p+1$ very ample, then the Koszul group $K_{p,2}(C,L)$ vanishes. In this article, we establish the conjecture in the extremal case, i.e.\ the case where the degree is exactly $2g+p+1-2h^1(C,L)-Cliff(C)$, subject to explicit genericity assumptions on $C$ and $L$. In particular, the gonality of $C$ is allowed to be arbitrary (in our cases $gon(C)=Cliff(C)+2$).
In this thesis we study singular curves on K3 surfaces. Let $\mathcal{B}_g$ denote the stack of polarised K3 surfaces of genus $g$ and set $p(g,k)=k^2(g-1)+1$. There is a stack $ \mathcal{T}^n_{g,k} \to \mathcal{B}_g$ with fibre over the polarised surface $(X,L)$ parametrising all unramified morphisms $f: C \to X$, birational onto their image, with $C$ an integral smooth curve of genus $ p(g,k)-n$ and $f_*C \sim kL$. One can think of $ \mathcal{T}^n_{g,k}$ as parametrising all singular curves on K3 surfaces such that the normalisation map is unramified (or equivalently such that the curve has "immersed" singularities). The stack $ \mathcal{T}^n_{g,k}$ comes with a natural moduli map $$η\; : \;\mathcal{T}^n_{g,k} \to \mathcal{M}_{p(g,k)-n}$$ to the Deligne-Mumford stack of curves, defined by forgetting the map to the K3 surface. We first show that $η$ is generically finite (to its image) on at least one component of $\mathcal{T}^n_{g,k} $, in all but finitely many values of $p(g,k)-n$. We also consider related questions about the Brill-Noether theory of singular curves on K3 surfaces as well as the surjectivity of twisted Gaussian maps on normalisations of singular curves. Lastly, we apply the deformation theory of $\mathcal{T}^n_{g,k}$ to a seemingly unrelated problem, namely the Bloch-Beilinson conjectures on the Chow group of points of K3 surfaces with a symplectic involution.
Generalizing the well-known Green Conjecture on syzygies of canonical curves, Green and Lazarsfeld formulated in 1986 the Secant Conjecture predicting that a line bundle L of sufficiently high degree on a curve has a non-linear p-syzygy if and only if L fails to be (p+1)-very ample. Via lattice theory for special K3 surfaces, Voisin's solution of the classical Green Conjecture and calculations on moduli stacks of pointed curves, we prove: (1) The Green-Lazarsfeld Secant Conjecture in various degree of generality, including its strongest possible form in the divisorial case in the universal Jacobian. (2) The Prym-Green Conjecture on the naturality of the resolution of a general Prym-canonical curve of odd genus.
In this article we consider moduli properties of singular curves on K3 surfaces. Let $\mathcal{B}_g$ denote the stack of primitively polarized K3 surfaces $(X,L)$ of genus $g$ and let $\mathcal{T}^n_{g,k} \to \mathcal{B}_g$ be the stack parametrizing tuples $[(f: C \to X, L)]$ with $f$ an unramified morphism which is birational onto its image, $C$ a smooth curve of genus $p(g,k)-n$ and $f_*C \in |kL|$. We show that the forgetful morphism $$η\; : \; \mathcal{T}^n_{g,k} \to \mathcal{M}_{p(g,k)-n}$$ is generically finite on one component, for all but finitely many values of $p(g,k)-n$. We further study the Brill--Noether theory of those curves parametrized by the image of $η$, and find a Wahl-type obstruction for a smooth curve with an unordered marking to have a nodal model on a K3 surface in such a way that the marking is the divisor over the nodes.
According to the Bloch-Beilinson conjectures, an automorphism of a K3 surface X that acts as the identity on the transcendental lattice should act trivially on CH^2(X). We discuss this conjecture for symplectic involutions and prove it in one third of all cases. The main point is to use special elliptic K3 surfaces and stable maps to produce covering families of elliptic curves on the generic K3 surface that are invariant under the involution.