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Sat, 04 Mar 2000 00:00:00 +0200Sat, 04 Mar 2000 00:00:00 +0200Plane curves of minimal degree with prescribed singularities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/757
We prove that there exists a positive \(\alpha\) such thatfor any integer \(\mbox{$d\ge 3$}\) and any topological types \(\mbox{$S_1,\dots,S_n$}\) of plane curve singularities, satisfying \(\mbox{$\mu(S_1)+\dots+\mu(S_n)\le\alpha d^2$}\), there exists a reduced irreducible plane curve of degree \(d\) with exactly \(n\) singular points of types \(\mbox{$S_1,\dots,S_n$}\), respectively. This estimate is optimal with respect to theexponent of \(d\). In particular, we prove that for any topological type \(S\) there exists an irreducible polynomial of degree \(\mbox{$d\le 14\sqrt{\mu(S)}$}\) having a singular point of type \(S\).Gert-Martin Greuel; Christoph Lossen; Eugenii Shustinarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/757Mon, 03 Apr 2000 00:00:00 +0200Moduli spaces for torsion free modules on curve singularities I
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/804
Gert-Martin Greuel; Gerhard Pfisterarticlehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/804Mon, 03 Apr 2000 00:00:00 +0200On an implementation of standard bases and syzygies in SINGULAR
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/461
Hubert Grassmann; Gert-Martin Greuel; Bernd Martin; W. Neumann; Gerhard Pfister; W. Pohl; Hans Schönemann; Thomas Siebertpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/461Mon, 03 Apr 2000 00:00:00 +0200SINGULAR version 1.2 User Manual
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/473
Gert-Martin Greuel; Gerhard Pfister; Hans Schönemannpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/473Mon, 03 Apr 2000 00:00:00 +0200Advances and improvements in the theory of standard bases and syzygies
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/475
Gert-Martin Greuel; Gerhard Pfisterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/475Mon, 03 Apr 2000 00:00:00 +0200Description of SINGULAR: A Computer Algebra System for Singularity Theory, Algebraic Geometry and Commutative Algebra
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/476
Gert-Martin Greuelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/476Mon, 03 Apr 2000 00:00:00 +0200The normalisation: a new algorithm, implementation and comparisons
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/775
Primary decomposition of an ideal in a polynomial ring over a field belongs to the indispensable theoretical tools in commutative algebra and algebraic geometry. Geometrically it corresponds to the decomposition of an affine variety into irreducible components and is, therefore, also an important geometric concept.The decomposition of a variety into irreducible components is, however, slightly weaker than the full primary decomposition, since the irreducible components correspond only to the minimal primes of the ideal of the variety, which is a radical ideal. The embedded components, although invisible in the decomposition of the variety itself, are, however, responsible for many geometric properties, in particular, if we deform the variety slightly. Therefore, they cannot be neglected and the knowledge of the full primary decomposition is important also in a geometric context.In contrast to the theoretical importance, one can find in mathematical papers only very few concrete examples of non-trivial primary decompositions because carrying out such a decomposition by hand is almost impossible. This experience corresponds to the fact that providing efficient algorithms for primary decomposition of an ideal I ae K[x1; : : : ; xn], K a field, is also a difficult task and still one of the big challenges for computational algebra and computational algebraic geometry.All known algorithms require Gr"obner bases respectively characteristic sets and multivariate polynomial factorization over some (algebraic or transcendental) extension of the given field K. The first practical algorithm for computing the minimal associated primes is based on characteristic sets and the Ritt-Wu process ([R1], [R2], [Wu], [W]), the first practical and general primary decomposition algorithm was given by Gianni, Trager and Zacharias [GTZ]. New ideas from homological algebra were introduced by Eisenbud, Huneke and Vasconcelos in [EHV]. Recently, Shimoyama and Yokoyama [SY] provided a new algorithm, using Gr"obner bases, to obtain the primary decompositon from the given minimal associated primes.In the present paper we present all four approaches together with some improvements and with detailed comparisons, based upon an analysis of 34 examples using the computer algebra system SINGULAR [GPS]. Since primary decomposition is a fairly complicated task, it is, therefore, best explained by dividing it into several subtasks, in particular, while sometimes only one of these subtasks is needed in practice. The paper is organized in such a way that we consider the subtasks separately and present the different approaches of the above-mentioned authors, with several tricks and improvements incorporated. Some of these improvements and the combination of certain steps from the different algorithms are essential for improving the practical performance.Wolfram Decker; Gert-Martin Greuel; Gerhard Pfister; Theo de Jongpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/775Mon, 03 Apr 2000 00:00:00 +0200On Schappert's characterization of strictlyunimodal plane curve singularities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/776
Y.A. Drozd; Gert-Martin Greuelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/776Mon, 03 Apr 2000 00:00:00 +0200On the classification of vector bundles on projective curves
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/777
We consider the "representation type" of the classification problem of vector bundles on a projective curve. We prove that this problem is always either finite, or tame, or wild and we completely describe those curves which are of finite, resp. tame, vector bundle type. We also give a complete list of indecomposable vector bundles for the finite and tame cases.Y.A. Drozd; Gert-Martin Greuelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/777Mon, 03 Apr 2000 00:00:00 +0200Some Aspects of Brieskorn's Mathematical Work
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/778
Gert-Martin Greuelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/778Mon, 03 Apr 2000 00:00:00 +0200Geometry of families of nodal curves on the blown-up projective plane
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/779
Let P2r be the projective plane blown up at r generic points. Denote by E0; E1; : : : ; Er the strict transform of a generic straight line on P2 and the exceptional divisors of the blown-up points on P2r respectively. We consider the variety Virr of all irreducible curves C with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility on families of reducible curves. For r ^ 9 we give the complete answer concerning the existence of nodal curves in Virr.Gert-Martin Greuel; Christop Lossen; Eugenii Shustinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/779Mon, 03 Apr 2000 00:00:00 +0200New asymptotics in the geometry of equisingular families of curves
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/781
Gert-Martin Greuel; Christop Lossen; Eugenii Shustinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/781Mon, 03 Apr 2000 00:00:00 +0200Castelnuvo Function, Zero-dimensional Schemes and Singular Plane Curves
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/782
We study families V of curves in P2(C) of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the expected dimension), respectively to be irreducible. For T-smoothness these conditions involve new invariants of curve singularities and are conjectured to be asymptotically proper, i.e., optimal up to a constant factor. To obtain the results, we study the Castelnuovo function, prove the irreducibility of the Hilbert scheme of zero-dimensional schemes associated to a cluster of infinitely near points of the singularities and deduce new vanishing theorems for ideal sheaves of zero-dimensional schemes in P2. Moreover, we give a series of examples of cuspidal curves where the family V is reducible, but where ss1(P2nC) coincides (and is abelian) for all C 2 V .Gert-Martin Greuel; Christoph Lossen; Eugenii Shustinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/782Mon, 03 Apr 2000 00:00:00 +0200Gröbner bases and algebraic geometry
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/783
After the notion of Gröbner bases and an algorithm for constructing them was introduced by Buchberger [Bu1, Bu2] algebraic geometers have used Gröbner bases as the main computational tool for many years, either to prove a theorem or to disprove a conjecture or just to experiment with examples in order to obtain a feeling about the structure of an algebraic variety. Nontrivial problems coming either from logic, mathematics or applications usually lead to nontrivial Gröbner basis computations, which is the reason why several improvements have been provided by many people and have been implemented in general purpose systems like Axiom, Maple, Mathematica, Reduce, etc., and systems specialized for use in algebraic geometry and commutative algebra like CoCoA, Macaulay and Singular. The present paper starts with an introduction to some concepts of algebraic geometry which should be understood by people with (almost) no knowledge in this field. In the second chapter we introduce standard bases (generalization of Gr"obner bases to non-well-orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions. The last chapter describes a new algorithm for computing the normalization of a reduced affine ring and gives an elementary introduction to singularity theory. Then we describe algorithms, using standard bases, to compute infinitesimal deformations and obstructions, which are basic for the deformation theory of isolated singularities. It is impossible to list all papers where Gr"obner bases have been used in local and global algebraic geometry, and even more impossible to give an overview about these contributions. We have, therefore, included only references to papers mentioned in this tutorial paper. The interested reader will find many more in the other contributions of this volume and in the literature cited there.Gert-Martin Greuel; Gerhard Pfisterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/783Mon, 03 Apr 2000 00:00:00 +0200Algorithmic ideal theory
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/784
Algebraic geometers have used Gröbner bases as the main computational tool for many years, either to prove a theorem or to disprove a conjecture or just to experiment with examples in order to obtain a feeling about the structure of an algebraic variety. Non-trivial mathematical problems usually lead to non-trivial Gröbner basis computations, which is the reason why several improvements and efficient implementations have been provided by algebraic geometers (for example, the systems CoCoA, Macaulay and SINGULAR). The present paper starts with an introduction to some concepts of algebraic geometry which should be understood by people with (almost) no knowledge in this field. In the second chapter we introduce standard bases (generalization of Gröbner bases to non-well-orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions. In the third chapter several algorithms for primary decomposition of polynomial ideals are presented, together with a discussion of improvements and preferable choices. We also describe a newly invented algorithm for computing the normalization of a reduced affine ring. The last chapter gives an elementary introduction to singularity theory and then describes algorithms, using standard bases, to compute infinitesimal deformations and obstructions, which are basic for the deformation theory of isolated singularities. It is impossible to list all papers where Gröbner basis have been used in local and global algebraic geometry, and even more impossible to give an overview about these contributions. We have, therefore, included only a few references to papers which contain interesting applications and which are not mentioned in this tutorial paper. The interested reader will find many more in the other contributions of this volume and in the literature cited there.Gert-Martin Greuel; Gerhard Pfisterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/784Mon, 03 Apr 2000 00:00:00 +0200Geometry of Equisingular Families of Curves
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/785
Singular algebraic curves, their existence, deformation, families (from the local and global point of view) attract continuous attention of algebraic geometers since the last century. The aim of our paper is to give an account of results, new trends and bibliography related to the geometry of equisingular families of algebraic curves on smooth algebraic surfaces over an algebraically closed field of characteristic zero. This theory is founded in basic works of Plücker, Severi, Segre, Zariski, and has tight links and finds important applications in singularity theory, topology of complex algebraic curves and surfaces, and in real algebraic geometry.Gert-Martin Greuel; Eugenii Shustinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/785Mon, 03 Apr 2000 00:00:00 +0200On moduli spaces of semiquasihomogeneous singularities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/805
Gert-Martin Greuel; Gerhard Pfisterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/805Mon, 03 Apr 2000 00:00:00 +0200Semicontinuity for representations of Cohen-Macaulay rings
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/807
Yurij Drozd; Gert-Martin Greuelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/807Mon, 03 Apr 2000 00:00:00 +0200Equianalytic and equisingular families of curves on surfaces
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/808
Gert-Martin Greuel; Christoph Lossenpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/808Mon, 03 Apr 2000 00:00:00 +0200Moduli Spaces of Semiquasihomogeneous Singularities with fixed Principal Part
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/810
Gert-Martin Greuel; Claus Hertling; Gerhard Pfisterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/810Mon, 03 Apr 2000 00:00:00 +0200Geometric quotients of unipotent group actions II
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/813
Gert-Martin Greuel; Gerhard Pfisterpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/813Mon, 03 Apr 2000 00:00:00 +0200