TY - GEN A1 - Martin, Bernd A1 - Süß, Hendrik T1 - Milnor algebras could be isomorphic to modular algebras Y1 - 2009 ER - TY - CHAP A1 - Siebert, Thomas T1 - Algorithms for the computation of free resolutions Y1 - 1999 ER - TY - GEN A1 - Süß, Hendrik A1 - Hausen, Jürgen T1 - The Cox ring of an algebraic variety with torus action T2 - Advances in Mathematics Y1 - 2010 U6 - https://doi.org/doi:10.1016/j.aim.2010.03.010 SN - 0001-8708 VL - 225 IS - 2 SP - 977 EP - 1012 ER - TY - GEN A1 - Süß, Hendrik A1 - Hausen, Jürgen A1 - Herppich, Elaine T1 - Multigraded Factorial Rings and Fano varieties with torus action T2 - Documenta Mathematica N2 - In a first result, we describe all finitely generated factorial algebras over an algebraically closed field of characteristic zero that come with an effective multigrading of complexity one by means of generators and relations. This enables us to construct systematically varieties with free divisor class group and a complexity one torus action via their Cox rings. For the Fano varieties of this type that have a free divisor class group of rank one, we provide explicit bounds for the number of possible deformation types depending on the dimension and the index of the Picard group in the divisor class group. As a consequence, one can produce classification lists for fixed dimension and Picard index. We carry this out expemplarily in the following cases. There are 15 non-toric surfaces with Picard index at most six. Moreover, there are 116 non-toric threefolds with Picard index at most two; nine of them are locally factorial, i.e. of Picard index one, and among these one is smooth, six have canonical singularities and two have non-canonical singularities. Finally, there are 67 non-toric locally factorial fourfolds and two one-dimensional families of non-toric locally factorial fourfolds. In all cases, we list the Cox rings explicitly. KW - Fano varieties KW - Cox ring KW - torus actions KW - graded ring Y1 - 2011 UR - http://www.math.uni-bielefeld.de/documenta/vol-16/vol-16.html SN - 1431-0635 VL - 16 SP - 71 EP - 109 ER - TY - GEN A1 - Süß, Hendrik A1 - Ilten, Nathan Owen T1 - Polarized complexity-1 T-varieties T2 - Michigan Mathematical Journal N2 - We describe polarized complexity-one T-varieties combinatorially in terms of so-called divisorial polytopes, and show how geometric properties of such a variety can be read off the corresponding divisorial polytope. We compare our description with other possible descriptions of polarized complexity-one T-varieties. We also describe how to explicitly find generators of affine complexity-one T-varieties. KW - torus actions KW - divisors Y1 - 2011 VL - 60 IS - 3 SP - 561 EP - 578 ER - TY - GEN A1 - Martin, Bernd A1 - Lavandoskyy, Victor T1 - A Symbolic Approach to Generation and Analysis of Finite Difference Schemes of Partial Differential Equations T2 - arXiv.org : (math-ph) KW - Computer Algebra KW - Differential Equation KW - Difference Schemes Y1 - 2010 UR - http://arxiv.org/pdf/1007.4443.pdf ER - TY - GEN A1 - Martin, Bernd A1 - Hirsch, Tobias T1 - Deformation with sections: Cotangent cohomology, flatnes conditions and modular germs T2 - arXiv.org : (math.CV) Y1 - 2003 UR - http://arxiv.org/pdf/math/0306281v1.pdf ER - TY - CHAP A1 - Hirsch, Tobias T1 - Computing the Integral Closure of an Ideal Using its Rees Algebra T2 - Computational commutative and non-commutative algebraic geometry, proceedings of the Nato Advanced Research Workshop on Computational Commutative and Non-Commutative Algebraic Geometry, Chisinau, Moldova, 6 - 11 June 2004 Y1 - 2004 SN - 1-586-03505-3 SP - 145 EP - 155 PB - IOS Press CY - Amsterdam [u.a.] ER - TY - CHAP A1 - Martin, Bernd A1 - Hirsch, Tobias T1 - Modular Strata of Deformation Functors T2 - Computational commutative and non-commutative algebraic geometry, proceedings of the Nato Advanced Research Workshop on Computational Commutative and Non-Commutative Algebraic Geometry, Chisinau, Moldova, 6 - 11 June 2004 Y1 - 2005 SN - 1-586-03505-3 SP - 156 EP - 166 PB - IOS Press CY - Amsterdam [u.a.] ER - TY - GEN A1 - Petersen, Lars A1 - Süß, Hendrik T1 - Torus invariant divisors T2 - Israel Journal of Mathematics N2 - Using the language of Altmann, Hausen and Süß, we describe invariant divisors on normal varieties X which admit an effective codimension one torus action. In this picture, X is given by a divisorial fan on a smooth projective curve Y. Cartierdivisors on X can be described by piecewise affine functions h on the divisorial fan S whereas Weil divisors correspond to certain zero and one-dimensional faces of it. Furthermore, we provide descriptions of the divisor class group and the canonical divisor. Global sections of line bundles O(D h ) will be determined by a subset of a weight polytope associatedto h, and global sections of specific line bundles on the underlying curve Y. Y1 - 2011 UR - 10.1007/s11856-011-0039-z SN - 0021-2172 SN - 1565-8511 VL - 182 IS - 1 SP - 481 EP - 504 ER - TY - GEN A1 - Martin, Bernd A1 - Pochekutov, Dimitry T1 - Discriminant and Singularities of Logarithmic Gauss Map, Examples and Application T2 - Journal of Siberian Federal University, Mathematik & Physics N2 - The study of hypersurfaces in a torus leads to the beautiful zoo of amoebas and their contours, whose possible configurations are seen from combinatorial data. There is a deep connection to the logarithmic Gauss map and its critical points. The theory has a lot of applications in many directions. In this report we recall basic notions and results from the theory of amoebas, show some connection to algebraic singularity theory and discuss some consequences from the well known classification of singularities to this subject. Moreover, we have tried to compute some examples using the computer algebra system SINGULAR and discuss different possibilities and their effectivity to compute the critical points. Here we meet an essential obstacle: Relevant examples need real or even rational solutions, which are found only by chance. We have tried to unify different views to that subject. KW - logarithmic Gauss map, singularities, discriminant, asymptotics, hypersurface amoeba Y1 - 2013 SN - 1997-1397 VL - 6 IS - 1 SP - 74 EP - 85 ER - TY - CHAP A1 - Martin, Bernd T1 - Computing versal deformations with SINGULAR Y1 - 1999 ER - TY - CHAP A1 - Martin, Bernd T1 - Modular lines for singularities of the T-seris KW - Unimodal Singularities KW - Deformations of singularities KW - Modulatity Y1 - 2007 ER - TY - JOUR A1 - Süß, Hendrik A1 - Altmann, Klaus A1 - Hausen, Jürgen T1 - Gluing affine torus actions via divisorial fans Y1 - 2008 ER - TY - JOUR A1 - Süß, Hendrik A1 - Ilten, Nathan Owen T1 - AG Codes from Polyhedral Divisors JF - arXiv.org : [math.AG] Y1 - 2008 UR - http://arxiv.org/pdf/0811.2696v2.pdf ER - TY - JOUR A1 - Martin, Bernd A1 - Grassmann, H. A1 - Greuel, G.-M. A1 - Neumann, W. A1 - Pfister, Gerhard A1 - Pohl, W. A1 - Schönemann, H. A1 - Siebert, Thomas T1 - On an implementation of standard bases and syzygies in SINGULAR Y1 - 1996 ER - TY - JOUR A1 - Martin, Bernd A1 - Hirsch, Tobias T1 - Deformations with section: Cotangent cohomology, flatness and modular subspaces Y1 - 2006 ER - TY - JOUR A1 - Martin, Bernd A1 - Aleksandrov, Aleksandr T1 - Gauss-Manin connection associated with bouquets of curves Y1 - 2003 ER - TY - JOUR A1 - Siebert, Thomas T1 - Recursive computation of free resolutions and a generalized Koszul complex Y1 - 2003 ER - TY - JOUR A1 - Martin, Bernd A1 - Süß, Hendrik T1 - Milnor algebras versus modular germs for unimodal hypersurface singularities KW - Hypersurface singularities KW - Milnor algebras KW - modular deformations KW - flatness strata Y1 - 2007 ER -