Related papers: Generic Syzygy Schemes

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…

Given a vector bundle $F$ on a variety $X$ and $W\subset H^0(F)$ such that the evaluation map $W\otimes \mathcal{O}_X\to F$ is surjective, its kernel $S_{F,W}$ is called generalized syzygy bundle. Under mild assumptions, we construct a…

Based on a recent result of Voisin [2001] we describe the last nonzero syzygy space in the linear strand of a canonical curve C of even genus g=2k lying on a K3 surface, as the ambient space of a k-2-uple embedded P^{k+1}. Furthermore the…

Let $Q_i(i=1,2)$ be $2g$ dimensional quadrics in $\mathbb{P}^{2g+1}$ and let $Y$ be the smooth intersection $Q_1\cap Q_2$. We associate the linear subspace in $Y$ with vector bundles on the hyperelliptic curve $C$ of genus $g$ by the left…

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.

We study the syzygies of canonical curves of genus $g\geq 3$ over an algebraically closed field $\mathbb{F}$ of characteristic $p>0$. We provide a new proof of generic Green's Conjecture for $p\geq\frac{g+4}{2}$. Using the techniques from…

This paper considers curves in Grassmannians which are themselves immersed in projective space by the Plucker map. It is shown that for a generic vector bundle of high enough degree, the image curve lies in a proper linear subvariety of…

Let $X\subset\mathbb{P}^r$ be an integral linearly normal variety and $R=k[x_0,\cdots,x_r]$ the coordinate ring of $\mathbb{P}^r$. It is known that the syzygies of $X$ contain some geometric information. In recent years the syzygies of…

In this paper we derive geometric consequences from the presence of a long strand of linear syzygies in the minimal free resolution of a closed scheme in projective space whose homogeneous ideal is generated by quadrics. These consequences…

We study the restrictions of rank 2 semistable vector bundles E on P^2 to conics. A Grauert-Mulich type theorem on the generic splitting is proven. The jumping conics are shown to have the scheme structure of a hypersurface J_{2} in P^5 of…

We define syzygy points of projective schemes, and introduce a program of studying their GIT stability. Then we describe two cases where we have managed to make some progress in this program, that of polarized K3 surfaces of odd genus, and…

We consider the Grassmannian $\mathbb{G}r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({\P^1},\O_{\P^1}(n))$. We define $\frak{X}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on $\P^1$…

In this work, we characterize matrices of linear forms and constant rank, demonstrating that, under some natural assumptions, they are always associated with a syzygy bundle that fits into a (partially linear) resolution. Furthermore, this…

We study smooth quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$ parameterising $6 \times 6$ skew-symmetric matrices of rank at most 4, not intersecting the Grassmannian $\mathbb{G}(1,5)$. Such surfaces correspond to…

This paper aims at generalizing some geometric properties of Grassmannians of finite dimensional vector spaces to the case of Grassmannnians of infinite dimensional ones, in particular for that of $k((z))$. It is shown that the Determinant…

We consider the problem of constructing matrices of linear forms of constant rank by focusing on the associated vector bundles on projective spaces. Important examples are given by the classical Steiner bundles, as well as some special…

Let E and F be vector bundles over a complex projective smooth curve X, and suppose that 0 -> E -> W -> F -> 0 is a nontrivial extension. Let G be a subbundle of F, and D an effective divisor on X. We give a criterion for the subsheaf G(-D)…

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…

We give the classification of globally generated vector bundles of rank $2$ on a smooth quadric surface with $c_1\le (2,2)$ in terms of the indices of the bundles, and extend the result to arbitrary higher rank case. We also investigate…

A projective scheme $X$ is called `quadratic' if $X$ is scheme-theoretically cut out by homogeneous equations of degree 2. Furthermore, we say $X$ satisfies `property $\textbf{N}_{2,p}$' if it is quadratic and the quadratic ideal has only…