Related papers: An effective bound for the gonality conjecture

We are concerned with the problem of the stability of the syzygy bundles associated to base point free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from Mark Green's vanishing…

Following N. Elkies ("ABC implies Mordell") we show that the abc conjecture of Masser-Oesterle implies an effective version of Siegel's theorem about integral points on algebraic curves, i.e. an upper bound for the S-integral points where…

We show that the boundary of a projectively compact Einstein manifold of dimension $n$ can be extended by a line bundle naturally constructed from the projective compactification. This extended boundary is such that its automorphisms can be…

We compute the A-infinity structure on the self-Ext algebra of the vector bundle $G$ over an elliptic curve of the form $G=\bigoplus_{i=1}^r P_i\oplus \bigoplus_{j=1}^s L_j$, where $(P_i)$ and $(L_j)$ are line bundles of degrees 0 and 1,…

We show the stability of certain syzygies of line bundles on curves, which we call transforms, and are kernels of the evaluation map on subspaces of the space of global sections. For the transforms constructed, we prove the existence of…

In response to Sanki-Vadnere, we present a short proof of the following theorem: a pair of simple curves on a hyperbolic surface whose complementary regions are disks has length at least half the perimeter of the regular right-angled…

Eisenbud and Harris conjectured in 1982 that an algebraic curve of high genus lies on a surface of low degree (which they proved for curves of very large degree). They observed constraints on the Hilbert function of a general hyperplane…

We prove explicit bounds on the number of lattice points on or near a convex curve in terms of geometric invariants such as length, curvature, and affine arclength. In several of our results we obtain the best possible constants. Our…

A relative Picard theory in the context of graded manifolds is introduced. A Berezinian calculus and a theory of connections over SUSY-curves are systematically developed, and used to prove a Gauss-Bonnet theorem for line bundles in that…

We study the dimension of loci of special line bundles on stable curves and for a fixed semistable multidegree. In case of total degree $d = g - 1$, we characterize when the effective locus gives a Theta divisor. In case of degree $g - 2$…

In this paper we compute the asymptotics of the metric on the line bundle over the moduli space of curves that arises when attempting to compute the archimedean height of the algebraic cycle $C-C^-$ in the jacobian of a smooth projective…

We compute explicitly the A-infinity structure on the Ext-algebra of the collection $({\mathcal O}_C, L)$, where $L$ is a line bundle of degree 1 on an elliptic curve $C$. The answer involves higher derivatives of Eisenstein series.

There are several notions of gonality for graphs. The divisorial gonality dgon(G) of a graph G is the smallest degree of a divisor of positive rank in the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the minimum degree…

A characterization of the symmetry algebra of the $n$th order ordinary differential equations (ODEs) with maximal symmetry and all third order linearizable ODEs is given. This is used to show that such an algebra $\mathfrak{g}$ determines…

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $\Delta(G)$ of $G$ is defined as the simple undirected graph whose vertex set ${\rm{V}}(G)$ consists…

In a first part of this paper, we prove constancy of the canonical graded Betti table among the smooth curves in linear systems on Gorenstein weak Fano toric surfaces. In a second part, we show that Green's canonical syzygy conjecture holds…

We formulate, and in some cases prove, three statements concerning the purity or, more generally the naturality of the resolution of various rings one can attach to a generic curve of genus g and a torsion point of order l in its Jacobian.…

The geometrical picture of gauge theories must be enlarged when a gauge potential ceases to behave like a connection, as it does in electroweak interactions. When the gauge group has dimension four, the vector space isomorphism between…

We consider the generic Green conjecture on syzygies of a canonical curve, and particularly the following reformulation thereof: {\it For a smooth projective curve $C$ of genus $g$ in characteristic 0, the condition ${\rm Cliff} C>l$ is…

Let $\Lambda$ be an artin algebra. We obtain that $\Lambda$ is syzygy-finite when the radical layer length of $\Lambda$ is at most two; as two consequences, we give a new upper bound for the dimension of the bounded derived category of the…