Gert-Martin Greuel

Cohen-Macaulay modules on hypersurface singularities II

0. Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 l. Periodic complexes and matrix factorizations . . . . . . . . . . . . . . . . . . . . 169 2. Construction of matrix factorizations and MCMs . . . . . . . . . . . . . . . . . . 173 3. Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4. MCMs on A~ and D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Introduction to Singularities and Deformations

Singularity theory is a field of intensive study in modern mathematics with fascinating relations to algebraic geometry, complex analysis, commutative algebra, representation theory, theory of Lie groups, topology, dynamical systems, and many more, and with numerous applications in the natural and technical sciences. This book presents the basic singularity theory of analytic spaces, including local deformation theory, and the theory of plane curve singularities. Plane curve singularities are a classical object of study, rich of ideas and applications, which still is in the center of current research and as such provides an ideal introduction to the general theory. Deformation theory is an important technique in many branches of contemporary algebraic geometry and complex analysis. This introductory text provides the general framework of the theory while still remaining concrete. In the first part of the book the authors develop the relevant techniques, including the Weierstras preparation theorem, the finite coherence theorem etc., and then treat isolated hypersurface singularities, notably the finite determinacy, classification of simple singularities and topological and analytic invariants. In local deformation theory, emphasis is laid on the issues of versality, obstructions, and equisingular deformations. The book moreover contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the mu-constant stratum which is based on deformations of the parameterization. Computational aspects of the theory are discussed as well. Three appendices, including basic facts from sheaf theory, commutative algebra, and formal deformation theory, make the reading self-contained. The material, which can be found partly in other books and partly in research articles, is presented from a unified point of view for the first time. It is given with complete proofs, new in many cases. The book thus can serve as source for special courses in singularity theory and local algebraic and analytic geometry.

SINGULAR: a computer algebra system for polynomial computations

SINGULAR is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, algebraic geometry, and singularity theory. SINGULAR’s main computational objects are polynomials, ideals and modules over a large variety of rings, including important non-commutative rings. SINGULAR features one of the fastest and most general implementations of various algorithms for computing standard resp. Gröbner bases. Furthermore, it provides multivariate polynomial factorization, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, numerical root–finding, visualization, and many more related functionalities.

Algorithmic Algebra and Number Theory

The aim of this article is to describe a computational approach to the study of the arithmetic of modular curves Xo(N) and to give applications of these computations.

Primary Decomposition: Algorithms and Comparisons

The Hilbert series and degree bounds play significant roles in computational invariant theory. In the modular case, neither of these tools is avrulable in general. In this article three results are obtruned, which provide partial remedies for these shortcomings. First, it is shown that the so-called extended Hilbert series, which can always be calculated by a MoHen type formula, yields strong constraints on the degrees of primary invariants. Then it is shown that for a trivial source module the (ordinary) Hilbert series coincides with that of a lift to characteristic 0 and can hence be calculated by MoHen’s formula. The last result is a generalization of Goobel’s degree bound to the case of monomial representations.

PLANE CURVES OF MINIMAL DEGREE WITH PRESCRIBED SINGULARITIES

Abstract. We prove that there exists a positive α such that for any integer d≥3 and any topological types S1,…,Sn of plane curve singularities, satisfying there exists a reduced irreducible plane curve of degree d with exactly n singular points of types S1,…,Sn, respectively. This estimate is optimal with respect to the exponent of d. In particular, we prove that for any topological type S there exists an irreducible polynomial of degree having a singular point of type S.

Constant Milnor Number Implies Constant Multiplicity for Quasihomogeneous Singularities.

We show that for certain topologically trivial deformations of an isolated hypersurface singularity the multiplicity does not change. This applies to all μ-constant “first order” deformations and to all μ-constant deformations of a quasihomogeneous singularity.

Equianalytic and equisingular families of curves on surfaces

SummaryWe consider flat families of reduced curves on a smooth surfaceS such that each memberC has the same number of singularities and each singularity has a fixed singularity type (up to analytic resp. topological equivalence). We show that these families are represented by a schemeH and give sufficient conditions for the smoothness ofH (atC). Our results improve previously known criteria for families with fixed analytic singularity type and seem to be quite sharp for curves in ℙ2 of small degree. Moreover, for families with fixed topological type this paper seems to be the first in which arbitrary singularities are treated.

Vector Bundles and Torsion Free Sheaves on Degenerations of Elliptic Curves

In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms, both methods are discussed here. Moreover, we include new proofs of some classical results about vector bundles on elliptic curves.

An Algebraic Approach for Proving Data Correctness in Arithmetic Data Paths

This paper proposes a new approach for proving arithmetic correctness of data paths in System-on-Chip modules. It complements existing techniques which are, for reasons of complexity, restricted to verifying only the control behavior. The circuit is modeled at the arithmetic bit level (ABL) so that our approach is well adapted to current industrial design styles for high performance data paths. Normalization at the ABL is combined with the techniques of computer algebra. We compute normal forms with respect to Grobner bases over rings i¾?/