Canonical Curves Research Articles

Gradients of odd theta functions

We show that a generic principally polarized abelian variety (ppav) is uniquely determined by its theta hyperplanes. These are the non-projectivized version of those studied by Caporaso and Sernesi (see math.AG/0204164), which in a sense are a generalization to ppavs of bitangents of plane quartics, and of hyperplanes tangent to a canonical curves of genus $g$ in $g-1$ points. More precisely, we show that, generically, the set of gradients of all odd theta functions at the point zero uniquely determines a ppav with level (4,8) structure. We also show that our map is an immersion of the moduli space of ppavs.

Geometric Syzygies of Elliptic Normal Curves and their Secant Varieties

We show that the linear syzygy spaces of elliptic normal curves, their secant varieties and of bielliptic canonical curves are spanned by geometric syzygies.

An interesting 0-cycle

The geometric and arithmetic properties of a smooth algebraic variety $X$ are reflected by the configuration of its subvarieties. A principal invariant of these are the Chow groups $\CH^p(X)$, defined to be the group of codimension-$p$ algebraic cycles modulo rational equivalence. For $p=1$ these groups are classical and well understood. For $p\dgeqq 2$ they are nonclassical in character and constitute a major area of study. In particular, it is generally difficult to decide whether a given higher codimension cycle is or is not rationally equivalent to zero. In their study of the moduli spaces of algebraic curves, C. Faber and R. Pandharipande introduced a canonical $0$-cycle $z_K$ on the product $X=Y\times Y$ of a curve $Y$ with itself. This cycle is of degree zero and Albanese equivalent to zero, and they asked whether or not it is rationally equivalent to zero. This is trivially the case when the genus $g=0,1,2$, and they proved that this is true when $g=3$. It is also the case when $Y$ is hyperelliptic or, conjecturally, when it is defined over a number field. We show that $z_K$ is not rationally equivalent when $Y$ is general and $g\dgeqq 4$. The proof is variational, and for it we introduce a new computational method using Shiffer variations. The condition $g\dgeqq 4$ enters via the property that the tangent lines to the canonical curve at two general points must intersect.

Divisors on [formula omitted] and the minimal resolution conjecture for points on canonical curves

We use geometrically defined divisors on moduli spaces of pointed curves to compute the graded Betti numbers of general sets of points on any nonhyperelliptic canonically embedded curve. This gives a positive answer to the Minimal Resolution Conjecture in the case of canonical curves. But we prove that the conjecture fails on curves of large degree. These results are related to the existence of theta divisors associated to certain stable vector bundles.

Canonical curves of genus eight

A non-tetragonal curve of genus 8 is a complete intersection of divisors in either $\mathbf{P}^2 \times \mathbf{P}^2$, a 6-dimensional weighted Grassmannian or the 8-dimensional Grassmannian.

Characterizing curves by their odd theta-characteristics

We show that a general canonical curve is uniquely determined by the finite set of hyperplanes cutting theta-characteristics on it. Geometrical and combinatorial properties of the moduli space of stable spin curves are proved, which play an essential role.

CANONICAL CURVES IN ℙ3

Let ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ be the Hilbert scheme of closed 1-dimensional subschemes of degree 6 and arithmetic genus 4 in $\mathbb{P}^3$. Let $H$ be the component of ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ whose generic point corresponds to a canonical curve, that is, a complete intersection of a quadric and a cubic surface in $\mathbb{P}^3$. Let $F$ be the vector space of linear forms in the variables $z_1, z_2, z_3, z_4$. Denote by $F_d$ the vector space of homogeneous forms of degree $d$. Set $X = \{(f_2,f_3)\}$ where $f_2 \in \mathbb{P}(F_2)$ is a quadric surface, and $f_3 \in \mathbb{P}(F_3/f_2 \cdot F)$ is a cubic modulo $f_2$. We have a rational map, $\sigma : X \cdots \rightarrow H$ defined by $(f_2,f_3) \mapsto f_2 \cap f_3$. It fails to be regular along the locus where $f_2$ and $f_3$ acquire a common linear component. Our main result gives an explicit resolution of the indeterminacies of $\sigma$ as well as of the singularities of $H$. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10, 14N15.

Material nano joins

Development and testing are described of a reduced model for the simulation of transient reactions in Ni/Al nanolaminates. The model relies on a simplified description of local atomic mixing rates, expressed in terms of an evolution equation for a dimensionless time scale that reflects the age of the mixed layer. The latter is obtained from a theoretical analysis of the quasi-1D evolution equation of a conserved scalar, which also yields canonical curves for the mean composition and its relevant moments as function of the local bilayer age. The reduced model is tested against results of the detailed model of Besnoin et al. [E. Besnoin, S. Cerutti, O.M. Knio, T.P. Weihs, J. Appl. Phys. 92 (2002) 5474–5481] for different scenarios, including self-propagating reactions, and ignition thresholds. Results show that the reduced model predictions are in good agreement with those obtained using the detailed model. By achieving order-of-magnitude increase in computation efficiency, the present model reduction formalism offers a path towards modeling transient reaction fronts in multiple space dimensions.

On canonical curves and osculating spaces Communicated by E.M. Friedlander

We study the geometry of a reduced canonical curve with a nondegenerate component. We prove that the other components are rational normal curves in a certain configuration. In addition, given a nonsingular point on a nondegenerate component, we analyze the relationship between the Weierstrass semigroup and the intersection divisors of the osculating spaces with the curve. We describe how these divisors vary and present an upper bound for their degrees. We study in detail the curves that attain this bound.

Green's conjecture for the generic r-gonal curve of genus g=3r-7

The syzygies of a generic canonical curve are expected to be as simple as possible for p≤(g−3)/2. We prove this result here for p≤(g−2)/3 only. The proof is carried out by considering infinitesimal deformations near a hyperelliptic curve.

Green's conjecture for the generic r-gonal curve of genus g≥3r−7

The syzygies of a generic canonical curve are expected to be as simple as possible for p≤(g−3)/2. We prove this result here for p≤(g−2)/3 only. The proof is carried out by considering infinitesimal deformations near a hyperelliptic curve.

On the number of moduli of extendable canonical curves

AbstractLet C ⊂ ℙg−1 be a canonical curve of genus g. In this article we study the problem of extendability of C, that is when there is a surface S ⊂ ℙg different from a cone and having C as hyperplane section. Using the work of Epema we give a bound on the number of moduli of extendable canonical curves. This for example implies that a family of large dimension of curves that are cover of another curve has general member nonextendable. Using a theorem of Wahl we prove the surjectivity of the Wahl map for the general k-gonal curve of genus g when k = 5, g ≥ 15 or k = 6, g ≥ 13 or k ≥ 7, g ≥ 12.

Canonical Curves and Varieties of Sums of Powers of Cubic Polynomials

In this note we show that the apolar cubic forms associated to codimension 2 linear sections of canonical curves of genus g≥11 are special with respect to their presentation as sums of cubes.

Generalized Weierstrass kernels on the intersection of two complex hypersurfaces

On plane algebraic curves the so-called Weierstrass kernel plays the same role of the Cauchy kernel on the complex plane. A straightforward prescription to construct the Weierstrass kernel has been known for more than one century. How can it be extended to the case of more general curves obtained from the intersection of hypersurfaces in an n-dimensional complex space? This problem is solved in this work in the case n=3. As an application, the correlation functions of bosonic string theories are constructed on a canonical curve of genus four.

The curve of “Prym canonical” Gauss divisors on a Prym theta divisor

The Gauss linear system on the theta divisor of the Jacobian of a nonhyperelliptic curve has two striking properties: 1) the branch divisor of the Gauss map on the theta divisor is dual to the canonical model of the curve; 2) those divisors in the Gauss system parametrized by the canonical curve

Gröbner Flags and Gorenstein Algebras

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s [les ] 2n points in general linear position in Pn, and the Koszul property of the ‘generic’ Artinian Gorenstein algebra of socle degree 3.

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.

Singularities of $2\Theta $-divisors in the jacobian

We consider the linear system 2Θ 0 | of second order theta functions over the Jacobian JC of a non-hyperelliptic curve C. A result by J. Fay says that a divisor D ∈ |2Θ 0 | contains the origin O ∈ JC with multiplicity 4 if and only if D contains the surface C - C = {O(p - q) | p, q ∈ C} C JC. In this paper we generalize Fay's result and some previous work by R.C. Gunning. More precisely, we describe the relationship between divisors containing O with multiplicity 6, divisors containing the fourfold C 2 - C 2 = {O(p + q - r - s) p, q, r, s ∈ C}, and divisors singular along C-C, using the third exterior product of the canonical space and the space of quadrics containing the canonical curve. Moreover we show that some of these spaces are equal to the linear span of Brill-Noether loci in the moduli space of semi-stable rank 2 vector bundles with canonical determinant over C, which can be embedded in |2Θ 0 |.

Non-classical trigonal curves

We study trigonal Gorenstein curves of arithmetic genus g ≥ 5 such that g⅓ has a base point, through realizing them as canonical curves lying over a cone. Using an explicit description of such curves we compute the dimension of their moduli spaces. We also investigate non-classical trigonal Gorenstein curves; in fact, we give a complete classification of such curves for arithmetic genus g = 5. More generally, we classify the non-classical curves when the characteristic of the constant field is g - 1, g - 2, or 2g - 3. In characteristic 2 we also solve the case g = 2n + 1.

On the geometry of canonical curves of odd genus

Fix an algebrically closed field k with char(k) = 0. Let C be a projective nonsingular curve of genus 5 defined over k, and let W 1 4 ⊂ Pic (C) be the subscheme of divisor classes of degree 4 and dimension 1. W 1 4 is a curve, which is irreducible and nonsingular of genus 11 if C is general, and can be identified with the singular locus of the theta divisorW4 ⊂ Pic(C). Under the Gauss rational map W 1 4 is mapped 2–1 onto a curve Γ ⊂ IP 4 which is nonsingular of degree 10 and genus 6. Γ is the set of vertices of the rank four quadrics containing the canonical image κ(C) ⊂ IP . The planes contained in the quadric QL, L ∈ W 1 4 , form the congruence (2-dimensional family) of 4-secant planes to κ(C). On each plane π of the congruence we have the set (κ(C) ∪ Γ) ∩ π consisting of 5 points. For a general choice of π these points are distinct and contained in a unique conic F ⊂ π which can be obtained as the locus of first order foci of the congruence, whereas the points ( κ(C) ∪ Γ ) ∩ π are the second order foci (see below for the definitions of first and second order foci). This geometrical configuration can be viewed as the first case of a whole series in two different ways. Firstly we can consider a sufficiently general curve C of genus g ≥ 5, and the locus W 1 g−1 ⊂ Pic g−1(C), which can be identified with the singular locus of the theta divisor Wg−1. W 1 g−1 has pure dimension g − 4; the projectivized tangent space at a point L ∈W 1 g−1\W 2 g−1 is the linear space vL ⊂ IP g−1 of dimension g− 5 vertex of the quadric QL of rank four containing the canonical curve κ(C), which is the projectivized tangent cone to Wg−1 at L. Each quadric QL has two rulings of IP g−3’s, and when L varies in W 1 g−1\W 2 g−1 they form a family of dimension g − 3. On each sufficiently general IP g−3 of the family there is a rational normal curve of first order foci. It has been shown in [CS] that the curve C can be recovered from this family, and this has been used to give a proof of Torelli’s theorem. Another extension of the genus 5 configuration can be introduced naturally, and it is the object of the present paper. Let’s consider, for any odd g = 2n+1, n ≥ 2, a sufficiently general projective irreducible nonsingular curve C of genus g, and the locusW 1 n+2 ⊂ Pic (C). By Brill-Noether theory it is known that W 1 n+2 is a nonsingular irreducible curve, whose genus has been computed by Kempf [K2] and by Pirola [P] (g = 11 for n = 2). For each L ∈W 1 n+2 the projectivized tangent cone to Wn+2 at L is a variety XL ⊂ IP , of degree n and dimension n+ 1, and XL = ⋃