Linear Equivalence Classes Research Articles

The rational cuspidal divisor class group of X0(N)

For any positive integer N, we completely determine the structure of the rational cuspidal divisor class group of X0(N), which is conjecturally equal to the rational torsion subgroup of J0(N). More specifically, for a given prime ℓ, we construct a rational cuspidal divisor Zℓ(d) for any non-trivial divisor d of N. Also, we compute the order of the linear equivalence class of Zℓ(d) and show that the ℓ-primary subgroup of the rational cuspidal divisor class group of X0(N) is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of Zℓ(d).

Moduli spaces of abstract and embedded Kummer varieties

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack [Formula: see text] of abstract Kummer varieties and the second one is the stack [Formula: see text] of embedded Kummer varieties. We will prove that [Formula: see text] is a Deligne-Mumford stack and its coarse moduli space is isomorphic to [Formula: see text], the coarse moduli space of principally polarized abelian varieties of dimension [Formula: see text]. On the other hand, we give a modular family [Formula: see text] of embedded Kummer varieties embedded in [Formula: see text], meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space [Formula: see text] of embedded Kummer surfaces and prove that it is obtained from [Formula: see text] by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: [Formula: see text] could be obtained from [Formula: see text] via a contraction for all [Formula: see text].

Divisor class groups of singular surfaces

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's theor em for the cubic ruled surface in P 3 . We apply these results to limit the possible curves that can be s et-theoretic complete intersection in P 3 in characteristic zero. On a nonsingular variety, the study of divisors and linear systems is classical. In fact the entire theory of curves and surfaces is dependent on this study of codimension one subvarieties and the linear and algebraic families in which they move. This theory has been generalized in two directions: the Weil divisors on a normal variety, taking codimension one subvarieties as prime divisors; and the Cartier divisors on an arbitrary scheme, based on locally principal codimension one subschemes. Most of the literature both in algebraic geometry and commutative algebra up to now has been limited to these kinds of divisors. More recently there have been good reasons to consider divisors on non-normal varieties. Jaffe (9) introduced the notion of an almost Cartier divisor, which is locally principal off a subset of codimension two. A theory of generalized divisors was proposed on curves in (14), and extended to any dimension in (15). The latter paper gave a complete description of the generalized divisors on the ruled cubic surface in P 3 . In this paper we extend that analysis to an arbitrary integra l surface X, explaining the group APicX of linear equivalence classes of almost Cartier divisors on X in terms of the Picard group of the normalization S of X and certain local data at the singular points of X. We apply these results to give limitations on the possible curves that can b e set-theoretic compete intersections in P 3 in characteristic zero In section 2 we explain our basic set-up, comparing divisors on a variety X to its normalization S. In Section 3 we prove a local isomorphism that computes the group of almost Cartier divisors at a singular point of X in terms of the Cartier divisors along the curve of singulari ties and its inverse image in the normalization. In Section 4 we derive some global exact sequences for the groups PicX, APicX, and PicS, which generalize the results of (15, §6) to arbitrary surfaces These results are particularly transparent for surfaces wi th ordinary singularities, meaning a double curve with a finite number of pinch points and triple points.

CANONICAL REPRESENTATIVES FOR DIVISOR CLASSES ON TROPICAL CURVES AND THE MATRIX–TREE THEOREM

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Gamma $ be a compact tropical curve (or metric graph) of genus $g$. Using the theory of tropical theta functions, Mikhalkin and Zharkov proved that there is a canonical effective representative (called a break divisor) for each linear equivalence class of divisors of degree $g$ on $\Gamma $. We present a new combinatorial proof of the fact that there is a unique break divisor in each equivalence class, establishing in the process an ‘integral’ version of this result which is of independent interest. As an application, we provide a ‘geometric proof’ of (a dual version of) Kirchhoff’s celebrated matrix–tree theorem. Indeed, we show that each weighted graph model $G$ for $\Gamma $ gives rise to a canonical polyhedral decomposition of the $g$-dimensional real torus $\mathrm{Pic}^g(\Gamma )$ into parallelotopes $C_T$, one for each spanning tree $T$ of $G$, and the dual Kirchhoff theorem becomes the statement that the volume of $\mathrm{Pic}^g(\Gamma )$ is the sum of the volumes of the cells in the decomposition.

Rank of divisors on tropical curves

We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, we confirm a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to the rank of D on the corresponding metric graph, and construct an algorithm for computing the rank of a divisor on a tropical curve.

Fixed-structure Gaussian process model

This article describes a method for modelling non-linear dynamic systems from measurement data. The method merges the linear local model blending approach in the velocity-based linearisation form with Bayesian Gaussian process (GP) modelling. The new Fixed-Structure GP (FSGP) model has a predetermined linear model structure with varying and probabilistic parameters represented by GP models. These models have several advantages for the modelling of local model parameters as they give us adequate results, even with small data sets. Furthermore, they provide a measure of the confidence in the prediction of the varying parameters and information about the dependence of the parameters on individual inputs. The FSGP model can be applied for the extended local linear equivalence class of non-linear systems. The obtained non-linear system model can be, for example, used for control-system design. The proposed modelling method is illustrated with a simple example of non-linear system modelling for control design.

On the classification of perfect codes: Extended side class structures

The two 1-error correcting perfect binary codes, C and C ′ are said to be equivalent if there exists a permutation π of the set of the n coordinate positions and a word d ̄ such that C ′ = π ( d ̄ + C ) . Hessler defined C and C ′ to be linearly equivalent if there exists a non-singular linear map φ such that C ′ = φ ( C ) . Two perfect codes C and C ′ of length n will be defined to be extended equivalent if there exists a non-singular linear map φ and a word d ̄ such that C ′ = φ ( d ̄ + C ) . Heden and Hessler, associated with each linear equivalence class an invariant L C and this invariant was shown to be a subspace of the kernel of some perfect code. It is shown here that, in the case of extended equivalence, the corresponding invariant will be the extension of the code L C . This fact will be used to give, in some particular cases, a complete enumeration of all extended equivalence classes of perfect codes.

Application of Varying Parameters Modelling with Gaussian Processes

Abstract Abstract This paper describes an application of the method for modelling nonlinear dynamic systems from measurement data. The method merges the linear local model blending approach in the velocity-based linearisation form with Bayesian Gaussian process modelling. The new Fixed-Structure Gaussian Process model has a predetermined linear model structure with varying and probabilistic parameters represented by Gaussian process models. These models have several advantages for the modelling of local model parameters as they give us adequate results, even with small data sets. Furthermore, they provide a measure of confidence in the prediction of the varying parameters and information about the dependence of the parameters on individual inputs. The Fixed-Structure Gaussian Process model can be used for the extended local linear equivalence class of nonlinear systems. The modelling method is applied to modelling of a semi-industrial gas-liquid plant that exhibits variable dynamics depending on its operating region.

Perfect codes as isomorphic spaces

Linear equivalence between perfect codes is defined. This definition gives the concept of general perfect 1-error correcting binary codes. These are defined as 1-error correcting perfect binary codes, with the difference that the set of errors is not the set of weight one words, instead any set with cardinality n and full rank is allowed. The side class structure defines the restrictions on the subspace of any general 1-error correcting perfect binary code. Every linear equivalence class will contain all codes with the same length, rank and dimension of kernel and all codes in the linear equivalence class will have isomorphic side class structures.

Arithmetic distance functions and height functions in diophantine geometry

In virtually all areas of Diophantine Geometry, the theory of height functions plays a crucial role. This theory associates to each (Cartier) divisor D on a projective variety V a function ho mapping the group of rational points of V to the real numbers. This function, which is defined up to a bounded function on V, has very nice functorial properties. For example, it is linear in D and depends only on the linear equivalence class of D. As a consequence, any relation between divisor classes, such as the theorem of the square for abelian varieties, will yield a corresponding relation for height functions. Further, it is possible to write the height function ho as a sum of local height functions 20(" ; v), where v ranges over the distinct absolute values of the given field. (These local heights are also known as logarithmic distance functions or Weil functions.) Each 2 o is defined away from the support of D, and gives a measure of the v-adic distance from the given point of V to the divisor D. In this paper we propose to define local height functions 2 x for every closed subscheme X of a projective variety V. As above, 2x(; v) will give a measure of the v-adic distance from a point of V to X. These functions will have many nice functorial properties; for example, 2xny will equal the minimum of 2x and hr. In this way, relations between closed subschemes (or equivalently, between ideal sheaves) will yield relations between local height functions. We will thus be able to convert geometric statements into arithmetic statements relatively painlessly. More generally, we will deal with the case that V is merely quasi-projective by defining a function 20v('; v) which will measure the v-adic distance to the "boundary" of V. We will then be able to assign to each closed subscheme X of V a local height function 2x which will be well defined up to addition of a multiple of 2~v. This extra generality is useful, for example, when one has a complete family of varieties and wishes to discard the "bad" fibers. As one application of the machinery that we have developed, we prove a quantitative version of the inverse function theorem. Thus given a finite map, we use local height functions to describe how far away from the ramification locus

A note on principal ideals and smooth curves

Let A be the coordinate ring of an affine piece of a smooth curve, V ̄ , defined over a field K. This paper investigates, for each positive integer n, the principal ideals of A of degree n. The main tool is the “Picard bundle” of V ̄ of degree n. For K of characteristic zero, V ̄ a “general” curve of genus g( V ̄ ) , and n < g( V ̄ ) , the Lang-Swinnerton-Dyer version of Modell's conjecture implies there are but finitely many linear equivalence classes of principal ideals of A of degree n. Conjectured conditions for V ̄ to be “general” are explicit and effectively computable.

Correspondencias divisoriales entre esquemas relativos

Let X Y be schemes over S . The divisorial correspondences between X, Y are define to be the linear equivalence classes of divisors on Xx S Y modulo the inverse images of the divisors on each factor . The main result is that the divisorial correspondences are a scheme over S whose geometric fibres are finitely generated abelian groups . A metric tensor on the divisorial correspondences is also given generalizing the trace metric for correspondences on curves, and it verifies a Castelnuovo inequality saying that it is positive definite modulo torsion.

The theorem of Torelli for singular curves

Let C be a compact (singular) curve embedded in a surface. Then C carries a canonical sheaf Ω \Omega which is locally free of rank 1. Moreover, C has a generalized Jacobian J which fits in an exact sequence ( ∗ ) 0 → F → J → A → 0 \begin{equation}\tag {$\ast $} 0 \to F \to J \to A \to 0\end{equation} of algebraic groups such that A is an abelian variety and F = ( C ∗ ) r × C s F = {({{\mathbf {C}}^\ast })^r} \times {{\mathbf {C}}^s} . Let C _ \underline {C} be the set of nonsingular points of C and let θ \theta = Zariski-closure of the image of ( C _ ) ( g − 1 ) (\underline {C})^{(g - 1)} in J. Then: Theorem. If C is irreducible and sections of Ω \Omega map C onto X in P g − 1 {P^{g - 1}} then the isomorphism class of J together with the translation class of the divisor θ \theta on J determine the isomorphism class of X. As a corollary, if ψ : C → X \psi :C \to X is an isomorphism (in which case we call C nonhyperelliptic) the above data determine the isomorphism class of C. I do not know if this remains true when C is hyperelliptic. It should be noted that the linear equivalence class of θ \theta is not enough to determine X. The principal idea of the proof is that of Andreotti, that is, to recover the curve as the dual of the branch locus of the Gauss map from θ \theta to P g − 1 {P^{g - 1}} ; however our arguments are usually analytic. The organization of this paper is as follows: In §1 we prove a stronger than usual version of Abel’s theorem for Riemann surfaces and in §2 we extend this theorem to apply to singular curves. In succeeding sections we construct the generalized Jacobian as a complex Lie group J and embed J in an analytic fibre bundle over A with projective spaces as fibre. This we use to endow J with the structure of an algebraic group. §7 contains a miscellany of facts about branch loci and dual varieties, and in §8 the main theorems are stated and proved. We should mention here that the variations on Abel’s theorem proved in this paper (1.2.4 and 3.0.1) were proved by Severi, at least in the special case corresponding to ordinary double points [12].