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Gorenstein threefold singularities with small resolutions via invariant theory of Weyl groups
 J. of Alg. Geom
, 1992
"... A fundamental new type of birational modification which first occurs in dimension three is the simple flip. This is a birational map Y �� � Y + which induces an isomorphism (Y −C) ∼ = (Y + −C +), where C and C + are smooth rational curves such that KY · C < 0 and KY + · C+> 0. (Y and Y + should be ..."
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Cited by 40 (5 self)
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A fundamental new type of birational modification which first occurs in dimension three is the simple flip. This is a birational map Y �� � Y + which induces an isomorphism (Y −C) ∼ = (Y + −C +), where C and C + are smooth rational curves such that KY · C < 0 and KY + · C+> 0. (Y and Y + should be allowed to have “terminal ” singularities.) Mori’s celebrated theorem [22] shows that these flips exist when numerically expected. A closely related type of modification is the simple flop. This has a similar definition, except that Y and Y + should be Gorenstein, with KY · C = KY + · C+ = 0. (This is more than an analogy: every flip has a branched double cover which is a flop, and this construction was used in Mori’s proof.) For both flips and flops, the curves C and C + can be contracted to points (in Y and Y +, respectively), yielding the same normal variety X. The birational map Y �� � Y + can thus be described in terms of the two contraction morphisms π: Y → X and
Multiple covers and the integrality conjecture for rational curves in CalabiYau threefolds
 J. Algebraic Geom
"... Abstract. We study the contribution of multiple covers of an irreducible rational curve C in a CalabiYau threefold Y to the genus 0 GromovWitten invariants in the following cases. 1. If the curve C has one node and satisfies a certain genericity condition, we prove that the contribution of multipl ..."
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Cited by 17 (3 self)
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Abstract. We study the contribution of multiple covers of an irreducible rational curve C in a CalabiYau threefold Y to the genus 0 GromovWitten invariants in the following cases. 1. If the curve C has one node and satisfies a certain genericity condition, we prove that the contribution of multiple covers of degree d is given by ∑ 1 n3 nd 2. For a smoothly embedded contractable curve C ⊂ Y we define schemes Ci for 1 ≤ i ≤ l where Ci is supported on C and has multiplicity i, the number l ∈ {1,...,6} being Kollár’s invariant “length”. We prove that the contribution of multiple covers of C of degree d is given by ∑ kd/n n nd 3 where ki is the multiplicity of Ci in its Hilbert scheme (and ki = 0 if i> l). In the latter case we also get a formula for arbitrary genus (Theorem 1.5). These results show that the curve C contributes an integer amount to the socalled instanton numbers that are defined recursively in terms of the GromovWitten invariants and are conjectured to be integers. 1. Motivation
The orbifold quantum cohomology of C 2 /Z3 and HurwitzHodge integrals
 Journal of Algebraic Geometry
"... Abstract. Let Z3 act on C 2 by nontrivial opposite characters. Let X = [C 2 /Z3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 GromovWitten potentials F X and F Y are equal after a change of variables — verifying the Crepant Resolution Conjec ..."
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Cited by 9 (1 self)
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Abstract. Let Z3 act on C 2 by nontrivial opposite characters. Let X = [C 2 /Z3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 GromovWitten potentials F X and F Y are equal after a change of variables — verifying the Crepant Resolution Conjecture for the pair (X, Y). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of independent interest. In a self contained Appendix, we derive closed formulas for these HurwitzHodge integrals. 1.
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.
Geometric transitions and n = 1 quiver theories
"... We construct N = 1 supersymmetric theories on worldvolumes of D5 branes wrapped around 2cycles of threefolds which are ADE fibrations over a plane. We propose large N duals as geometric transitions involving blowdowns of two cycles and blowups of threecycles. This yields exact predictions for a l ..."
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Cited by 6 (0 self)
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We construct N = 1 supersymmetric theories on worldvolumes of D5 branes wrapped around 2cycles of threefolds which are ADE fibrations over a plane. We propose large N duals as geometric transitions involving blowdowns of two cycles and blowups of threecycles. This yields exact predictions for a large class of N = 1 supersymmetric gauge systems including U(N) gauge theories with two adjoint matter fields deformed by superpotential terms, which arise in ADE fibered geometries with nontrivial monodromies. August
AuslanderReiten theory revisited
 In Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep
, 2008
"... Abstract. We recall several results in AuslanderReiten theory for finitedimensional algebras over fields and orders over complete local rings. Then we introduce ncluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained. ..."
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Cited by 2 (2 self)
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Abstract. We recall several results in AuslanderReiten theory for finitedimensional algebras over fields and orders over complete local rings. Then we introduce ncluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.