K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

Arithmetically Cohen Macaulay embeddings of reducible curves

Here we study the arithmetically Cohen—Macaulay non-special embeddings of reducible connected complete algebraic curves. We allow to take “general” the irreducible components and the line bundle (with fixed multidegree) and in this case we obtain sharp results.     

Oriented matroids and complete-graph embeddings on surfaces

We provide a link between topological graph theory and pseudoline arrangements from the theory of oriented matroids. We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface. Each element of the pair keeps the information of the oriented matroid we started with. We call a simple pseudoline arrangement triangular, when the cells in the cell decomposition of the projective plane are 2-colorable and when one color class of cells consists of triangles only. Precisely for triangular pseudoline arrangements, one element of the image pair of f is a triangular complete-graph embedding on a surface. We obtain all triangular complete-graph embeddings on surfaces this way, when we extend the definition of triangular complete pseudoline arrangements in a natural way to that of triangular curve arrangements on surfaces in which each pair of curves has a point in common where they cross. Thus Ringel's results on the triangular complete-graph embeddings can be interpreted as results on curve arrangements on surfaces. Furthermore, we establish the relationship between 2-colorable curve arrangements and Petrie dual maps. A data structure, called intersection pattern is provided for the study of curve arrangements on surfaces. Finally we show that an orientable surface of genus g admits a complete curve arrangement with at most 2g+1 curves in contrast to the non-orientable surface where the number of curves is not bounded.     

Sums of residues on algebraic surfaces and application to coding theory

In this paper, we study residues of differential 2-forms on a smooth algebraic surface over an arbitrary field and give several statements about sums of residues. Afterwards, using these results, we construct algebraic-geometric codes which are an extension to surfaces of the well-known differential codes on curves. We also study some properties of these codes and extend to them some known properties for codes on curves.     

The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface

For a simple complete ideal ℘ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincaré series P_℘, that gathers in a unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to ℘. This paper is devoted to prove that P_℘ is a rational function giving an explicit expression for it.     

Testing Graph Isotopy on Surfaces

We investigate the following problem: Given two embeddings G ₁ and G ₂ of the same abstract graph G on an orientable surface S, decide whether G ₁ and G ₂ are isotopic; in other words, whether there exists a continuous family of embeddings between G ₁ and G ₂. We provide efficient algorithms to solve this problem in two models. In the first model, the input consists of the arrangement of G ₁ (resp., G ₂) with a fixed graph cellularly embedded on S; our algorithm is linear in the input complexity, and thus, optimal. In the second model, G ₁ and G ₂ are piecewise-linear embeddings in the plane, minus a finite set of points; our algorithm runs in O(n ^3/2logn) time, where n is the complexity of the input. The graph isotopy problem is a natural variation of the homotopy problem for closed curves on surfaces and on the punctured plane, for which algorithms have been given by various authors; we use some of these algorithms as a subroutine. As a by-product, we reprove the following mathematical characterization, first observed by Ladegaillerie (Topology 23:303–311, 1984): Two graph embeddings are isotopic if and only if they are homotopic and congruent by an oriented homeomorphism.     

Direct limits of Cohen–Macaulay rings

Let A be a direct limit of a direct system of Cohen–Macaulay rings. In this paper, we describe the Cohen–Macaulay property of A. Our results indicate that A is not necessarily Cohen–Macaulay. We show A is Cohen–Macaulay under various assumptions. As an application, we study Cohen–Macaulayness of non-affine normal semigroup rings.     

Embedding compact surfaces into the 3-dimensional Euclidean space with maximum symmetry

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space R~3 are easier to feel by human's intuition. We give the maximum order of finite group actions on(R~3, Σ) among all possible embedded closed/bordered surfaces with given geometric/algebraic genus greater than 1 in R~3. We also identify the topological types of the bordered surfaces realizing the maximum order, and find simple representative embeddings for such surfaces.     

Intersection theory over quantum ruled surfaces

In this paper, we will extend several results on intersection theory over commutative ruled surfaces to quantum ruled surfaces. Typically, we define the fiber of a closed point, the quasi-section, and the quasi-canonical divisor on a quantum rules surface, and study how these “curves” on a quantum ruled surface intersect with each other.     

General blowups of ruled surfaces

In this note we study blowups of algebraic surfaces of Kodaira dimension κ = - ∞ at general points, their embeddings and secant varieties of the embedded surfaces.     

Effective invariants of braid monodromy

In this paper we construct new invariants of algebraic curves based on (not necessarily generic) braid monodromies. Such invariants are effective in the sense that their computation allows for the study of Zariski pairs of plane curves. Moreover, the Zariski pairs found in this work correspond to curves having conjugate equations in a number field, and hence are not distinguishable by means of computing algebraic coverings. We prove that the embeddings of the curves in the plane are not homeomorphic. We also apply these results to the classification problem of elliptic surfaces.

     

Maximal rank and minimal generation of some parametric varieties

In this paper, continuing work of the second author (J. Pure Appl. Algebra 155 (2001) 77) for rational curves, we address the problem of computing the generators of the ideal of an irreducible parametric variety V to the computation of the generators of the ideal of a suitable finite set of points on V. In particular, we consider the case of general parametric surfaces and threefolds and of general parametric surfaces represented by polynomials with base points.     

K-Theory of curves over number fields

We consider the algebraic K-groups with coefficients of smooth curves over number fields. We give a proof of the Quillen-Lichtenbaum conjecture at the prime 2 and prove explicit corank formulas for the algebraic K-groups with divisible coefficients. At odd primes these formulas assume the Bloch-Kato conjecture, at the prime 2 the formulas hold nonconjecturally.     

Maximal and Cohen–Macaulay projective monomial curves

In this paper we continue the investigation of Cohen–Macaulay projective monomial curves begun in [Les Reid, Leslie G. Roberts, Non-Cohen–Macaulay projective monomial curves, J. Algebra 291 (2005) 171–186]. In the process we introduce maximal curves. Cohen–Macaulay curves are maximal, but not conversely. We show that the number of all curves of degree d that are Cohen–Macaulay grows exponentially, but not as fast as the total number of curves, and also that maximal curves of degree d with sufficiently large embedding dimension relative to d are Cohen–Macaulay.     

Abeliants and their application to an elementary construction of Jacobians

The abeliant is a polynomial rule which to each n×n by n+2 array with entries in a commutative ring with unit associates an n×n matrix with entries in the same ring. The theory of abeliants, first introduced in an earlier paper of the author, is simplified and extended here. Now let J be the Jacobian of a nonsingular projective algebraic curve defined over an algebraically closed field. With the aid of the theory of abeliants we obtain explicit defining equations for J and its group law.     

Opers and theta functions

We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of semistable vector bundles over an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli space of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit coordinates on the moduli space. The finiteness results give low-dimensional parametrizations of Jacobians (in for generic curves), described by 2Θ functions or second logarithmic derivatives of theta.We interpret the Klein and Wirtinger maps in terms of opers on X. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The Wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the Szegö kernel. This generalizes the Wirtinger projective connections associated to theta characteristics, and the associated Klein bidifferentials.     

Second order homological obstructions on real algebraic manifolds

Let Y be a compact nonsingular real algebraic set whose homology classes (over Z/2) are represented by Zariski closed subsets. It is well known that every smooth map from a compact smooth manifold to Y is unoriented bordant to a regular map. In this paper, we show how to construct smooth maps from compact nonsingular real algebraic sets to Y not homotopic to any regular map starting from a nonzero homology class of Y of positive degree. We use these maps to obtain obstructions to the existence of local algebraic tubular neighborhoods of algebraic submanifolds of Rn and to study some algebro-homological properties of rational real algebraic manifolds.     

Elliptic solutions of nonlinear integrable equations and related topics

The theory of elliptic solitons for the Kadomtsev-Petviashvili (KP) equation and the dynamics of the corresponding Calogero-Moser system is integrated. It is found that all the elliptic solutions for the KP equation manifest themselves in terms of Riemann theta functions which are associated with algebraic curves admitting a realization in the form of a covering of the initial elliptic curve with some special properties. These curves are given in the paper by explicit formulae. We further give applications of the elliptic Baker-Akhiezer function to generalized elliptic genera of manifolds and to algebraic 2-valued formal groups.Dedicated to the memory of J.-L. Verdier     

Multiple supersingular elliptic fibers on elliptic surfaces

Elliptic surfaces over an algebraically closed field in characteristic p>0 with multiple supersingular elliptic fibers, that is, multiple fibers of a supersingular elliptic curve, are investigated. In particular, it is shown that for an elliptic surface with q=g+1 and a supersingular elliptic curve as a general fiber, where q is the dimension of an Albanese variety of the surface and g is the genus of the base curve, the multiplicities of the multiple supersingular elliptic fibers are not divisible by p². As an application of this result, the structure of false hyperelliptic surfaces is discussed on this basis.