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69
Mirror Symmetry and Rational Curves on Quintic Threefolds: A Guide for Mathematicians
 J. AM. MATH. SOC
, 1993
"... We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new qexpansion principle for functions on the ..."
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Cited by 107 (10 self)
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We give a mathematical account of a recent string theory calculation which predicts the number of rational curves on the generic quintic threefold. Our account involves the interpretation of Yukawa couplings in terms of variations of Hodge structure, a new qexpansion principle for functions on the moduli space of CalabiYau manifolds, and the “mirror symmetry” phenomenon recently observed by string theorists.
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 + sho ..."
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Cited by 71 (10 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
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 38 (15 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.
On K3 surfaces admitting finite nonsymplectic group actions
 J. Math. Sci. Univ. Tokyo
"... Abstract. For a pair (X,G) of a complex K3 surface X and its finite automorphism group G, we call the value I(X,G): =  Im(G → Aut(H2,0(X)))  the transcendental value and the Euler number ϕ(I(X,G)) of I(X,G) the transcendental index. This paper classifies the pairs (X,G) with the maximal transcend ..."
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Cited by 31 (5 self)
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Abstract. For a pair (X,G) of a complex K3 surface X and its finite automorphism group G, we call the value I(X,G): =  Im(G → Aut(H2,0(X)))  the transcendental value and the Euler number ϕ(I(X,G)) of I(X,G) the transcendental index. This paper classifies the pairs (X,G) with the maximal transcendental index 20 and the pair (X,G) with I(X,G) = 40 up to isomorphisms. We also determine the set of transcendental values and apply this to determine the set of global canonical indices of complex projective threefolds with only canonical singularities and with numerically trivial canonical Weil divisor. Let X be a K3 surface, that is, a simply connected smooth projective complex surface with a nowhere vanishing holomorphic two form. We denote by SX, TX and ωX the Néron Severi lattice, the transcendental lattice and a nowhere vanishing holomorphic two form of X. We denote the multi
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 30 (6 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
Genus zero GopakumarVafa invariants of contractible curves
"... Abstract. A version of the DonaldsonThomas invariants of a CalabiYau threefold is proposed as a conjectural mathematical definition of the GopakumarVafa invariants. These invariants have a local version, which is verified to satisfy the required properties for contractible curves. This provides a ..."
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Cited by 27 (1 self)
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Abstract. A version of the DonaldsonThomas invariants of a CalabiYau threefold is proposed as a conjectural mathematical definition of the GopakumarVafa invariants. These invariants have a local version, which is verified to satisfy the required properties for contractible curves. This provides a new viewpoint on the computation of the local GromovWitten invariants of contractible curves by Bryan, Leung, and the author. 1 Introduction. Let X be a CalabiYau threefold, β ∈ H2(X,Z), and g a nonnegative integer. The GopakumarVafa invariants n g β were first introduced as an integervalued index arising from Dbranes and M2branes wrapping holomorphic curves in
BIRATIONAL PROPERTIES OF PENCILS OF DEL Pezzo Surfaces of degree 1 and 2
, 2000
"... In this paper we study the birational rigidity problem for smooth Mori fibrations on del Pezzo surfaces of degree 1 and 2. For degree 1 we obtain a complete description of rigid and nonrigid cases. ..."
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Cited by 18 (1 self)
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In this paper we study the birational rigidity problem for smooth Mori fibrations on del Pezzo surfaces of degree 1 and 2. For degree 1 we obtain a complete description of rigid and nonrigid cases.
Projective morphisms according to Kawamata
, 1983
"... X is a projective 3fold with canonical singularities, k = C; the terminology will be explained in 0.8 below. Theorem 0.0 (on projective morphisms) Let D ∈ PicX be nef, and suppose that aD − KX is nef and big for some a ∈ Z with a ≥ 1. Then ..."
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Cited by 17 (0 self)
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X is a projective 3fold with canonical singularities, k = C; the terminology will be explained in 0.8 below. Theorem 0.0 (on projective morphisms) Let D ∈ PicX be nef, and suppose that aD − KX is nef and big for some a ∈ Z with a ≥ 1. Then
Crepant resolution of trihedral singularities
 Proceedings of the Japan Academy 70
, 1994
"... The purpose of this paper is to construct a crepant resolution of quotient singularities by finite subgroups of SL(3, C) of certain type, and prove that each Euler number of the minimal model is equal to the number of conjugacy classes. The problem of finding a nice resolution of quotient singularit ..."
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Cited by 16 (0 self)
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The purpose of this paper is to construct a crepant resolution of quotient singularities by finite subgroups of SL(3, C) of certain type, and prove that each Euler number of the minimal model is equal to the number of conjugacy classes. The problem of finding a nice resolution of quotient singularities by finite subgroups
Threefold divisorial contractions to singularities of higher indices
"... Abstract. We complete the explicit study of a threefold divisorial contraction whose exceptional divisor contracts to a point, by treating the case where the point downstairs is a singularity of index n≥2. We prove that if this singularity is of type cA/n then any such contraction is a suitable wei ..."
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Cited by 12 (2 self)
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Abstract. We complete the explicit study of a threefold divisorial contraction whose exceptional divisor contracts to a point, by treating the case where the point downstairs is a singularity of index n≥2. We prove that if this singularity is of type cA/n then any such contraction is a suitable weighted blowup; and that if otherwise then the discrepancy is 1/n with a few exceptions. Every such exception has an example. Some exceptions allow the discrepancy to be arbitrarily large, but any contraction in this case is described as a weighted blowup of a singularity of type cD/2 embedded into a cyclic quotient of a smooth fivefold. The erratum to the previous paper [14] is attached.