Advances in Mathematics, ISSN 0001-8708, 08/2019, Volume 352, pp. 372 - 405

We introduce tautological systems defined by prehomogeneous actions of reductive algebraic groups. If the complement of the open orbit is a linear free divisor...

Mixed Hodge modules | Tautological systems | Linear free divisors | PERIOD INTEGRALS | POLYNOMIALS | MATHEMATICS | SYMMETRY | DUALITY | HYPERGEOMETRIC SYSTEMS | Differential equations

Mixed Hodge modules | Tautological systems | Linear free divisors | PERIOD INTEGRALS | POLYNOMIALS | MATHEMATICS | SYMMETRY | DUALITY | HYPERGEOMETRIC SYSTEMS | Differential equations

Journal Article

Mathematical Research Letters, ISSN 1073-2780, 2017, Volume 24, Issue 4, pp. 1023 - 1042

A characterization of freeness for plane curves in terms of the Hilbert function of the associated Milnor algebra is given as well as many new examples of...

Rational cuspidal curve | Jacobian ideal | Milnor algebra | Free divisor | free divisor | MATHEMATICS | OPEN SURFACES | NUMBER | rational cuspidal curve

Rational cuspidal curve | Jacobian ideal | Milnor algebra | Free divisor | free divisor | MATHEMATICS | OPEN SURFACES | NUMBER | rational cuspidal curve

Journal Article

Annali di Matematica Pura ed Applicata (1923 -), ISSN 0373-3114, 8/2016, Volume 195, Issue 4, pp. 1393 - 1403

On a metric graph, we introduce the notion of a free divisor as a replacement for the notion of a base point-free complete linear system on a curve. By means...

Graph | 05C99 | Mathematics, general | Mathematics | Clifford index | 14H51 | Free divisor | 14T05 | MATHEMATICS | MATHEMATICS, APPLIED | TROPICAL CURVES | RANK

Graph | 05C99 | Mathematics, general | Mathematics | Clifford index | 14H51 | Free divisor | 14T05 | MATHEMATICS | MATHEMATICS, APPLIED | TROPICAL CURVES | RANK

Journal Article

Mathematische Zeitschrift, ISSN 0025-5874, 6/2013, Volume 274, Issue 1, pp. 249 - 261

We investigate differential systems occurring in the study of particular non-isolated singularities, the so-called linear free divisors. We obtain a duality...

Holonomic dual | 32S40 | Spectral numbers | Mathematics, general | Birkhoff problem | 34M35 | Mathematics | Frobenius manifold | Linear free divisors | Brieskorn lattice | MATHEMATICS

Holonomic dual | 32S40 | Spectral numbers | Mathematics, general | Birkhoff problem | 34M35 | Mathematics | Frobenius manifold | Linear free divisors | Brieskorn lattice | MATHEMATICS

Journal Article

Journal of Singularities, ISSN 1949-2006, 2013, Volume 7, pp. 253 - 274

J. Singul. 7 (2013), 253-274 We describe two situations where adding the adjoint divisor to a divisor D with smooth normalization yields a free divisor. Both...

Versal deformation | Prehomogeneous vector space | Free divisor | Isolated singularity | Adjoint divisor | Stable map | Mathematics - Algebraic Geometry

Versal deformation | Prehomogeneous vector space | Free divisor | Isolated singularity | Adjoint divisor | Stable map | Mathematics - Algebraic Geometry

Journal Article

Journal of Algebra, ISSN 0021-8693, 05/2016, Volume 453, pp. 221 - 248

Let S be a standard graded Artinian algebra over a field k. We identify constraints on the Hilbert function of S which are imposed by the hypothesis that S...

Exact pair of zero divisors | Determinantal ring | Segre embedding | Linear resolution | Totally acyclic complex | Compressed level algebra | Generic points in projective space | Matrix factorization | Pfaffians | Tate resolution | Totally reflexive module | CODIMENSION | RINGS | FREE RESOLUTIONS | IDEALS | COMPRESSED ALGEBRAS | MATHEMATICS | DIMENSION | TOTALLY REFLEXIVE MODULES | ARBITRARY SOCLE-VECTORS | Algebra

Exact pair of zero divisors | Determinantal ring | Segre embedding | Linear resolution | Totally acyclic complex | Compressed level algebra | Generic points in projective space | Matrix factorization | Pfaffians | Tate resolution | Totally reflexive module | CODIMENSION | RINGS | FREE RESOLUTIONS | IDEALS | COMPRESSED ALGEBRAS | MATHEMATICS | DIMENSION | TOTALLY REFLEXIVE MODULES | ARBITRARY SOCLE-VECTORS | Algebra

Journal Article

Topology and its Applications, ISSN 0166-8641, 2012, Volume 159, Issue 2, pp. 437 - 449

We apply previous results on the representations of solvable linear algebraic groups to construct a new class of free divisors whose complements are K ( π , 1...

Exceptional orbit varieties | (Modified) Cholesky-type factorizations | Block representations | Relative invariants | Cohomology of Milnor fibers | Linear free divisors | Solvable linear algebraic groups | Cohomology of complements | Eilenberg–Mac Lane spaces | Eilenberg-mac lane spaces | Cohomology of milnor fibers | Eilenberg-Mac Lane spaces | MATHEMATICS, APPLIED | DISCRIMINANTS | SPACES | MATHEMATICS

Exceptional orbit varieties | (Modified) Cholesky-type factorizations | Block representations | Relative invariants | Cohomology of Milnor fibers | Linear free divisors | Solvable linear algebraic groups | Cohomology of complements | Eilenberg–Mac Lane spaces | Eilenberg-mac lane spaces | Cohomology of milnor fibers | Eilenberg-Mac Lane spaces | MATHEMATICS, APPLIED | DISCRIMINANTS | SPACES | MATHEMATICS

Journal Article

Communications in Algebra, ISSN 0092-7872, 01/2019, Volume 47, Issue 1, pp. 424 - 449

Kaplansky Zero Divisor Conjecture states that if G is a torsion-free group and is a field, then the group ring contains no zero divisor and Kaplansky Unit...

Kaplansky zero divisor conjecture | 16S34 | Group ring | 20C07 | Kaplansky unit conjecture | torsion-free group | MATHEMATICS | PRODUCT | RINGS | SUBSETS

Kaplansky zero divisor conjecture | 16S34 | Group ring | 20C07 | Kaplansky unit conjecture | torsion-free group | MATHEMATICS | PRODUCT | RINGS | SUBSETS

Journal Article

Mathematische Nachrichten, ISSN 0025-584X, 08/2014, Volume 287, Issue 11-12, pp. 1383 - 1393

In this paper, we study the minimal free resolution of a nondegenerate projective variety X⊂Pr when X is contained in a variety Y of minimal degree as a...

Primary: 14N05 | 13D02 | Divisor of a rational normal scroll | Minimal free resolution | MATHEMATICS | VARIETIES

Primary: 14N05 | 13D02 | Divisor of a rational normal scroll | Minimal free resolution | MATHEMATICS | VARIETIES

Journal Article

Annales de l'Institut Fourier, ISSN 0373-0956, 2015, Volume 65, Issue 3, pp. 1251 - 1300

We introduce a method for obtaining new classes of free divisors from representations V of connected linear algebraic groups G where dim G = dim V, with V...

Prehomogeneous vector spaces | Exceptional orbit varieties | Solvable algebraic groups | Block representations | Cholesky-type factorizations | Free divisors | Determinantal varieties | Infinite-dimensional solvable Lie algebras | Linear free divisors | Pfaffian varieties | prehomogeneous vector spaces | CODIMENSION | DISCRIMINANTS | free divisors | exceptional orbit varieties | solvable algebraic groups | IDEALS | PREHOMOGENEOUS VECTOR-SPACES | MATHEMATICS | linear free divisors | VANISHING TOPOLOGY | infinite-dimensional solvable Lie algebras | LEGACY | block representations | FAMILIES | DEFORMATIONS | determinantal varieties

Prehomogeneous vector spaces | Exceptional orbit varieties | Solvable algebraic groups | Block representations | Cholesky-type factorizations | Free divisors | Determinantal varieties | Infinite-dimensional solvable Lie algebras | Linear free divisors | Pfaffian varieties | prehomogeneous vector spaces | CODIMENSION | DISCRIMINANTS | free divisors | exceptional orbit varieties | solvable algebraic groups | IDEALS | PREHOMOGENEOUS VECTOR-SPACES | MATHEMATICS | linear free divisors | VANISHING TOPOLOGY | infinite-dimensional solvable Lie algebras | LEGACY | block representations | FAMILIES | DEFORMATIONS | determinantal varieties

Journal Article

Advances in Mathematics, ISSN 0001-8708, 08/2015, Volume 281, pp. 1242 - 1273

In this paper we prove that the Bernstein–Sato polynomial of any free divisor for which the D[s]-module D[s]hs admits a Spencer logarithmic resolution...

Logarithmic differential operators | Spencer resolutions | Bernstein–Sato polynomials | Logarithmic connections | Lie–Rinehart algebras | Free divisors | Lie-Rinehart algebras | Bernstein-Sato polynomials | COMPLEMENT | HOLONOMIC SYSTEMS | LINEAR FREE DIVISORS | RHAM COMPLEXES | MATHEMATICS | COHOMOLOGY | LOGARITHMIC COMPARISON THEOREM | Mathematics - Algebraic Geometry

Logarithmic differential operators | Spencer resolutions | Bernstein–Sato polynomials | Logarithmic connections | Lie–Rinehart algebras | Free divisors | Lie-Rinehart algebras | Bernstein-Sato polynomials | COMPLEMENT | HOLONOMIC SYSTEMS | LINEAR FREE DIVISORS | RHAM COMPLEXES | MATHEMATICS | COHOMOLOGY | LOGARITHMIC COMPARISON THEOREM | Mathematics - Algebraic Geometry

Journal Article

Journal of Algebraic Combinatorics, ISSN 0925-9899, 5/2013, Volume 37, Issue 3, pp. 523 - 543

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an...

Armendariz map | Comaximal graph | Annihilating-ideal graph | Convex and Discrete Geometry | Graph invariants | Mathematics | Order, Lattices, Ordered Algebraic Structures | Group Theory and Generalizations | Combinatorics | Computer Science, general | Zero-divisor graph | Nilpotent-free semigroup | MATHEMATICS | Invisibility | Universities and colleges

Armendariz map | Comaximal graph | Annihilating-ideal graph | Convex and Discrete Geometry | Graph invariants | Mathematics | Order, Lattices, Ordered Algebraic Structures | Group Theory and Generalizations | Combinatorics | Computer Science, general | Zero-divisor graph | Nilpotent-free semigroup | MATHEMATICS | Invisibility | Universities and colleges

Journal Article

manuscripta mathematica, ISSN 0025-2611, 11/2018, Volume 157, Issue 3, pp. 361 - 385

Using a general stable vector bundle, we give an embedding $$\alpha _Y$$ αY of the compactified Jacobian $$\bar{J}(Y)$$ J¯(Y) of an integral nodal curve Y into...

Geometry | Topological Groups, Lie Groups | 14H60 | Calculus of Variations and Optimal Control; Optimization | Mathematics, general | Algebraic Geometry | Mathematics | Number Theory | MATHEMATICS | GENERALIZED PARABOLIC BUNDLES | TORSION-FREE SHEAVES | VECTOR-BUNDLES

Geometry | Topological Groups, Lie Groups | 14H60 | Calculus of Variations and Optimal Control; Optimization | Mathematics, general | Algebraic Geometry | Mathematics | Number Theory | MATHEMATICS | GENERALIZED PARABOLIC BUNDLES | TORSION-FREE SHEAVES | VECTOR-BUNDLES

Journal Article

Turkish Journal of Mathematics, ISSN 1300-0098, 2016, Volume 40, Issue 4, pp. 824 - 831

SLx be the free semilattice on a finite nonempty set X. There exists an undirected graph Gamma(SLx) associated with SLx whose vertices are the proper subsets...

Hamiltonian graph | Clique number | Finite free semilattice | Zero-divisor graph | Perfect graph | Domination number | MATHEMATICS | zero-divisor graph | domination number | clique number | perfect graph

Hamiltonian graph | Clique number | Finite free semilattice | Zero-divisor graph | Perfect graph | Domination number | MATHEMATICS | zero-divisor graph | domination number | clique number | perfect graph

Journal Article

Mathematical Research Letters, ISSN 1073-2780, 2017, Volume 24, Issue 5, pp. 1477 - 1496

We prove that any divisor as in the title must be normal crossing.

Normal crossing | Logarithmic derivation | Free divisor | Lie algebra | Quasihomogeneity | free divisor | MATHEMATICS | logarithmic derivation | normal crossing | quasihomogeneity | DERIVATIONS | Mathematics - Algebraic Geometry

Normal crossing | Logarithmic derivation | Free divisor | Lie algebra | Quasihomogeneity | free divisor | MATHEMATICS | logarithmic derivation | normal crossing | quasihomogeneity | DERIVATIONS | Mathematics - Algebraic Geometry

Journal Article

Journal of Commutative Algebra, ISSN 1939-0807, 4/2013, Volume 5, Issue 1, pp. 17 - 47

We present several methods to construct or identify families of free divisors such as those annihilated by many Euler vector fields, including binomial free...

Discriminant | Saito matrix | Euler | Free divisor | Vector field | Binomial | MATHEMATICS | binomial | discriminant | Euler vector field

Discriminant | Saito matrix | Euler | Free divisor | Vector field | Binomial | MATHEMATICS | binomial | discriminant | Euler vector field

Journal Article

Publications of the Research Institute for Mathematical Sciences, ISSN 0034-5318, 09/2013, Volume 49, Issue 3, pp. 393 - 412

We study a natural generalization of transversally intersecting smooth hypersurfaces in a complex manifold: hypersurfaces whose components intersect in a...

Hilbert-Samuel polynomial | Jacobian ideal | Free divisor | Normal crossing divisor | Logarithmic derivations | free divisor | LIE-ALGEBRAS | MATHEMATICS | normal crossing divisor | logarithmic derivations

Hilbert-Samuel polynomial | Jacobian ideal | Free divisor | Normal crossing divisor | Logarithmic derivations | free divisor | LIE-ALGEBRAS | MATHEMATICS | normal crossing divisor | logarithmic derivations

Journal Article

Journal of Number Theory, ISSN 0022-314X, 10/2014, Volume 143, pp. 109 - 124

For any positive integer n, let n=q1⋯qs be the prime factorization of n with q1≥⋯≥qs>1. A positive integer n is said to be ordinary if the smallest positive...

Divisors | Ordinary integers | Extraordinary integers | Square-free integers | MATHEMATICS

Divisors | Ordinary integers | Extraordinary integers | Square-free integers | MATHEMATICS

Journal Article

Communications in Algebra, ISSN 0092-7872, 02/2018, Volume 46, Issue 2, pp. 887 - 925

Kaplansky's zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field , the group ring [G] has no zero...

zero divisor | zero unit graph | 16S34 | torsion-free group | group ring | Kaplansky's unit conjecture | Divisor graph | 20C07 | Kaplansky's zero divisor conjecture | Kaplansky’s unit conjecture | Kaplansky’s zero divisor conjecture | MATHEMATICS | GROUP-RINGS | CONJECTURE | Torsion | Group theory

zero divisor | zero unit graph | 16S34 | torsion-free group | group ring | Kaplansky's unit conjecture | Divisor graph | 20C07 | Kaplansky's zero divisor conjecture | Kaplansky’s unit conjecture | Kaplansky’s zero divisor conjecture | MATHEMATICS | GROUP-RINGS | CONJECTURE | Torsion | Group theory

Journal Article

Journal of Algebra and its Applications, ISSN 0219-4988, 09/2015, Volume 14, Issue 7

An infinite family of irreducible homogeneous free divisors in K[x, y, z] is constructed. Indeed, we identify sets of monomials X such that the general...

syzygy | Free divisor | Saito matrix | MATHEMATICS | MATHEMATICS, APPLIED

syzygy | Free divisor | Saito matrix | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

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