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26
Introduction to the log minimal model program for log canonical pairs
, 2009
"... We describe the foundation of the log minimal model program for log canonical ..."
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Cited by 57 (17 self)
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We describe the foundation of the log minimal model program for log canonical
Syzygies, multigraded regularity and toric varieties
, 2006
"... Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B1,...,Bℓ on X and m1,...,mℓ ∈ N, consider the line bundle L: = B m1 1 ⊗···⊗Bmℓ ℓ. We give conditions on the mi which guarantee that the ide ..."
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Cited by 19 (4 self)
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Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B1,...,Bℓ on X and m1,...,mℓ ∈ N, consider the line bundle L: = B m1 1 ⊗···⊗Bmℓ ℓ. We give conditions on the mi which guarantee that the ideal of X in P(H0 (X, L) ∗ ) is generated by quadrics and that the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.
Introduction to the toric Mori theory
 MICHIGAN MATH. J
, 2003
"... The main purpose of this paper is to give a simple and noncombinatorial proof to the toric Mori theory. Here, the toric ..."
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Cited by 18 (6 self)
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The main purpose of this paper is to give a simple and noncombinatorial proof to the toric Mori theory. Here, the toric
Multiplication maps and vanishing theorems for toric varieties
 MATH. Z
, 2006
"... We use multiplication maps to give a characteristicfree approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems. ..."
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Cited by 15 (3 self)
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We use multiplication maps to give a characteristicfree approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.
BOUNDS ON FAKE WEIGHTED PROJECTIVE SPACE
, 2005
"... A fake weighted projective space X is a Qfactorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (λ0,...,λn). We see how the singularities of P(λ0,...,λn) influence the singularities of X, and how the weights bound the number of po ..."
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Cited by 13 (2 self)
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A fake weighted projective space X is a Qfactorial toric variety with Picard number one. As with weighted projective space, X comes equipped with a set of weights (λ0,...,λn). We see how the singularities of P(λ0,...,λn) influence the singularities of X, and how the weights bound the number of possible fake weighted projective spaces for a fixed dimension. Finally, we present an upper bound on the ratios λj / ∑ λi if we wish X to have only terminal (or canonical) singularities.
Volume and lattice points of reflexive simplices
 Discrete Comput. Geom
"... Let X be a Gorenstein toric Fano variety with class number one, e.g., a weighted projective space with Gorenstein singularities. We give sharp upper bounds on the anticanonical degree (−KX) dim X of X and on the anticanonical degree −KX.C of torusinvariant integral curves C on X, and we completely ..."
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Cited by 8 (0 self)
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Let X be a Gorenstein toric Fano variety with class number one, e.g., a weighted projective space with Gorenstein singularities. We give sharp upper bounds on the anticanonical degree (−KX) dim X of X and on the anticanonical degree −KX.C of torusinvariant integral curves C on X, and we completely characterize the cases of equality. For this we prove sharp upper bounds on the volume and the number of lattice points on edges of reflexive simplices. Furthermore we state a kind of BlaschkeSantaló inequality. These results are derived from bounds on the denominators of
Equivariant completions of toric contraction morphisms
 Tohoku Math. J
"... Abstract. We treat equivariant completions of toric contraction morphisms as an application of the toric Mori theory. For this purpose, we generalize the toric Mori theory for nonQfactorial toric varieties. So, our theory seems to be quite different from Reid’s original combinatorial toric Mori th ..."
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Cited by 8 (3 self)
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Abstract. We treat equivariant completions of toric contraction morphisms as an application of the toric Mori theory. For this purpose, we generalize the toric Mori theory for nonQfactorial toric varieties. So, our theory seems to be quite different from Reid’s original combinatorial toric Mori theory. We also explain various examples of nonQfactorial contractions, which imply that the Qfactoriality plays an important role in the Minimal Model Program. Thus, this paper completes the foundations of the toric Mori theory and show us a new aspect of the Minimal Model Program. Contents
The nef cone of toroidal compactifications of A4
 J. London Math. Soc
"... The moduli space Ag of principally polarised abelian gfolds is a quasiprojective variety. It has a natural projective compactification, the Satake ..."
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Cited by 7 (3 self)
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The moduli space Ag of principally polarised abelian gfolds is a quasiprojective variety. It has a natural projective compactification, the Satake
Mirror symmetry and tropical geometry
 Ph.D. thesis, Universität des Saarlandes
, 2007
"... Using tropical geometry we propose a mirror construction for monomial degenerations of CalabiYau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for CalabiYau hypersurfaces and by Batyrev and Borisov for CalabiYau complete intersections. We apply ..."
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Cited by 6 (0 self)
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Using tropical geometry we propose a mirror construction for monomial degenerations of CalabiYau varieties in toric Fano varieties. The construction reproduces the mirror constructions by Batyrev for CalabiYau hypersurfaces and by Batyrev and Borisov for CalabiYau complete intersections. We apply the construction to Pfaffian examples and recover the mirror given by Rødland for the degree 14 CalabiYau threefold in P 6 defined by the Pfaffians of a general linear 7 × 7 skewsymmetric matrix. We provide the necessary background knowledge entering into the tropical mirror construction such as toric geometry, Gröbner bases, tropical geometry, Hilbert schemes and deformations. The tropical approach yields an algorithm which we illustrate in a series of explicit examples. Contents