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57
Triangulated categories of singularities and Dbranes in LandauGinzburg models
 Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240–262
, 2005
"... Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend ..."
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Cited by 95 (4 self)
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Dedicated to the blessed memory of Andrei Nikolaevich Tyurin – adviser and friend
Varieties of sums of powers
, 1998
"... The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results (cf. [Sylvester 1851], [Hilbert 1888], [Dixon, Stuart 1906]) and some more recent ..."
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Cited by 22 (3 self)
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The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results (cf. [Sylvester 1851], [Hilbert 1888], [Dixon, Stuart 1906]) and some more recent results of Mukai (cf. [Mukai 1992]) are presented together with new results for the cases (n, d) = (3, 8), (4, 2), (5, 3). In the last case the variety of sums of 8 powers of a general cubic form is a Fano 5fold of index 1 and degree 660.
Orientifolds of Gepner Models
, 2004
"... We systematically construct and study Type II Orientifolds based on Gepner models which have N = 1 supersymmetry in 3+1 dimensions. We classify the parity symmetries and construct the crosscap states. We write down the conditions that a configuration of rational branes must satisfy for consistency ( ..."
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Cited by 20 (3 self)
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We systematically construct and study Type II Orientifolds based on Gepner models which have N = 1 supersymmetry in 3+1 dimensions. We classify the parity symmetries and construct the crosscap states. We write down the conditions that a configuration of rational branes must satisfy for consistency (tadpole cancellation and rank constraints) and spacetime supersymmetry. For certain cases, including Type IIB orientifolds of the quintic and a two parameter model, one can find all solutions in this class. Depending on the parity, the number of vacua can be large, of the order of 10 10 −10 13. For other models, it is hard to find all solutions but special solutions can be found — some of them are chiral. We also make comparison with the large volume regime and obtain a perfect match. Through this study, we find a number of new features of Type II orientifolds, including the structure of moduli space and the change
Local Rings Of Finite CohenMacaulay Type
, 1997
"... this paper was partially supported by the National Science Foundation. ..."
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Cited by 14 (4 self)
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this paper was partially supported by the National Science Foundation.
Ascent of Finite CohenMacaulay Type
 J. Algebra
, 1999
"... this paper we prove the other direction. Specifically, we prove the following theorem. ..."
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Cited by 12 (7 self)
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this paper we prove the other direction. Specifically, we prove the following theorem.
Finite Gorenstein representation type implies simple singularity
 Adv. Math
"... Abstract. Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive Rmodules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated sin ..."
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Cited by 9 (4 self)
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Abstract. Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive Rmodules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive Rmodule is free.
RANK 2 ARITHMETICALLY COHENMACAULAY BUNDLES ON A NONSINGULAR CUBIC SURFACE
, 2005
"... Abstract. Rank 2 indecomposable arithmetically CohenMacaulay bundles E on a nonsingular cubic surface X in P 3 are classified, by means of the possible forms taken by the minimal graded free resolution of E over P 3. The admissible values of the Chern classes of E are listed and the vanishing locus ..."
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Cited by 9 (2 self)
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Abstract. Rank 2 indecomposable arithmetically CohenMacaulay bundles E on a nonsingular cubic surface X in P 3 are classified, by means of the possible forms taken by the minimal graded free resolution of E over P 3. The admissible values of the Chern classes of E are listed and the vanishing locus of a general section of E is studied. Properties of E such as slope (semi) stability and simplicity are investigated; the number of relevant families is computed together with their dimension. 1.
Graña; Vector bundles on G(1, 4) without intermediate cohomology
"... A known result by Horrocks (see [H]) characterizes the line bundles on a projective space as the only indecomposable vector bundles without intermediate cohomology. This result has been generalized by Ottaviani (see [O1], [O2]) to quadrics and Grassmannians. ..."
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Cited by 8 (2 self)
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A known result by Horrocks (see [H]) characterizes the line bundles on a projective space as the only indecomposable vector bundles without intermediate cohomology. This result has been generalized by Ottaviani (see [O1], [O2]) to quadrics and Grassmannians.
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 8 (7 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
A SPLITTING CRITERION FOR RANK 2 BUNDLES ON A GENERAL SEXTIC THREEFOLD
, 2004
"... Abstract. In this paper we show that on a general sextic hypersurface X ⊂ P 4, a rank 2 vector bundle E splits if and only if h 1 (E(n)) = 0 for any n ∈ Z. We get thus a characterization of complete intersection curves in X. 1. ..."
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Cited by 6 (0 self)
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Abstract. In this paper we show that on a general sextic hypersurface X ⊂ P 4, a rank 2 vector bundle E splits if and only if h 1 (E(n)) = 0 for any n ∈ Z. We get thus a characterization of complete intersection curves in X. 1.