On Koszul property of the homogeneous coordinate ring of a curve

The following corollary has been added: for general tetragonal curve $C$ of genus $g\ge 9$ the homogeneous coordinate ring of $C$ defined by the line bundle $K(-T)$, where $K$ is the canonical class, $T$ is the tetragonal series, is Koszul. Also some misprints are corrected.Comment: 17 pages, Late

Rolling Factors Deformations and Extensions of Canonical Curves

A tetragonal canonical curve is the complete intersection of two divisors on a scroll. The equations can be written in `rolling factors' format. For such homogeneous ideals we give methods to compute infinitesimal deformations. Deformations can be obstructed. For the case of quadratic equations on the scroll we derive explicit base equations. They are used to study extensions of tetragonal curves.Comment: 38 pp., plain Te

Gonality, apolarity and hypercubics

We show that any Fermat hypercubic is apolar to a trigonal curve, and vice versa. We show also that the Waring number of the polar hypercubic associated to a tetragonal curve of genus $g$ is at most $\lceil 3/2g - 7/2\rceil$, and for a large class of them is at most $4/3g - 3$.Comment: 9 pages, to appear in the Bulletin of the London Mathematical Societ

A genus six cyclic tetragonal reduction of the Benney equations

A reduction of Benney’s equations is constructed corresponding to Schwartz–Christoffel maps associated with a family of genus six cyclic tetragonal curves. The mapping function, a second kind Abelian integral on the associated Riemann surface, is constructed explicitly as a rational expression in derivatives of the Kleinian σ-function of the curve

Abelian functions associated with a cyclic tetragonal curve of genus six

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y^4 = x^5 + λ[4]x^4 + λ[3]x^3 + λ[2]x^2 + λ[1]x + λ[0]. We construct Abelian functions using the multivariate sigma-function associated with the curve, generalizing the theory of theWeierstrass℘-function. We demonstrate that such functions can give a solution to the KP-equation, outlining how a general class of solutions could be generated using a wider class of curves. We also present the associated partial differential equations satisfied by the functions, the solution of the Jacobi inversion problem, a power series expansion for σ(u) and a new addition formula

A combinatorial interpretation for Schreyer's tetragonal invariants

Schreyer has proved that the graded Betti numbers of a canonical tetragonal curve are determined by two integers b(1) and b(2), associated to the curve through a certain geometric construction. In this article we prove that in the case of a smooth projective tetragonal curve on a toric surface, these integers have easy interpretations in terms of the Newton polygon of its defining Laurent polynomial. We can use this to prove an intrinsicness result on Newton polygons of small lattice width

The trigonal construction in the ramified case

Abstract To every double cover ramified in two points of a general trigonal curve of genus g$g$, one can associate an étale double cover of a tetragonal curve of genus g+1$g+1$. We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties. This extends Recillas' trigonal construction to covers ramified in two points.To every double cover ramified in two points of a general trigonal curve of genus g, one can associate an étale double cover of a tetragonal curve of genus g + 1. We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties. This extends Recillas' trigonal construction to covers ramified in two points.Peer Reviewe

Triplanar Model for the Gap and Penetration Depth in YBCO

YBaCuO_7 is a trilayer material with a unit cell consisting of a CuO_2 bilayer with a CuO plane of chains in between. Starting with a model of isolated planes coupled through a transverse matrix element, we consider the possibility of intra as well as interplane pairing within a nearly antiferromagnetic Fermi liquid model. Solutions of a set of three coupled BCS equations for the gap exhibit orthorhombic symmetry with s- as well as d-wave contributions. The temperature dependence and a-b in plane anisotropy of the resulting penetration depth is discussed and compared with experiment.Comment: To appear in Physical Review B1 01Mar97; 12 pages with 10 figures; RevTeX+eps