Showing papers on "Toric variety published in 1992"
TL;DR: A new way to use marked sequences to encode permutations is developed, which provides a transparent explanation of the unimodality of Eulerian numbers and their isotypic refinements.
75 citations
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TL;DR: In this paper, an erratum that corrects an error in the proof of Proposition 4.3 in my paper "The Homogeneous Coordinate Ring of a Toric Variety" is presented.
Abstract: This submission consists of two papers: 1) an erratum that corrects an error in the proof of Proposition 4.3 in my paper "The Homogeneous Coordinate Ring of a Toric Variety", and 2) the original (unchanged) version of the paper, published in 1995. The original paper introduced the homogeneous coordinate ring of a toric variety (now called the total coordinate ring or Cox ring) and gave a quotient construction. The paper also studied sheaves on a toric variety, and in Section 4 described its automorphism group. The error in the proof of Proposition 4.3 resulted from the faulty assumption that a certain set of graded endomorphisms forms a ring; rather, it is a monoid under composition. The erratum notes this error and gives a correct proof of the proposition.
56 citations
TL;DR: In this article, the authors give a geometric description of special T- invariant Cartier divisors of a compact toric variety and use their description in order to develop a formula for the calculation of the rank of the Picard group.
Abstract: First we give a geometric description of special T- invariant Cartier divisors of a compact toric variety. We show that these divisors always exist and use their description in order to develop a formula for the calculation of the rank of the Picard group of a compact toric variety. Moreover we give an example of two combinatorially equivalent fans Σ1 and Σ2 such that} % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!
$X_{{\Sigma}_1}$
is projective with a non- trivial Picard group and
% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A!
$X_{{\Sigma}_1}$
is non- projective with a surprisingly trivial Picard group. Furthermore, we show that Σ2 is a fan which cannot be spanned by any topological sphere which is the union of (d − l)- polytopes such that the polytopes correspond exactly to the full dimensional cones of Σ.
31 citations
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TL;DR: In this article, the authors describe the second cohomology Brauer group with coefficients in the sheaf of units of any toric variety, which is the so-called cohomological Brauer Group of $X$.
Abstract: Toric varieties are a special class of rational varieties defined by equations of the form {\it monomial = monomial}. For a good brief survey of the history and role of toric varieties see [10]. Any toric variety $X$ contains a cover by affine open sets described in terms of arrangements (called fans) of convex bodies in $\Bbb R^r$. The coordinate rings of each of these affine open sets is a graded ring generated over the ground field by monomials. As a consequence, toric varieties provide a good context in which cohomology can be calculated. The purpose of this article is to describe the second \'etale cohomology group with coefficients in the sheaf of units of any toric variety $X$. This is the so-called cohomological Brauer group of $X$.
14 citations
TL;DR: In this article, the universal Grobner basis of a unimodular toric ideal is characterized and the main new result in this paper is the characterization of the universal GBS of a toric basis.
Abstract: This partially expository note extends and refines our earlier work on Grobner bases of toric varieties [14]. It is closely related to the theory of A-hypergeometric functions due to Gel′fand Graev, Kapranov and Zelevinsky (see § 6). The main new result in this note is the characterization of the universal Grobner basis of a unimodular toric ideal (see § 5).
14 citations
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TL;DR: In this article, the authors show how one might compute the ''etale cohomology groups'' in degrees with coefficients in the sheaf of units of a toric variety with integral coefficients.
Abstract: The purpose of this article is to show how one might compute the \'etale cohomology groups $H^p_{\acute{e}t}(X,G_m)$ in degrees $p=0$, $1$ and $2$ of a toric variety $X$ with coefficients in the sheaf of units. The method is to reduce the computation down to the problem of diagonalizing a matrix with integral coefficients. The procedure outlined in this article has been fully implemented by the author as a program written in the ``C'' programming language.
11 citations
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TL;DR: In particular, Danilov et al. as discussed by the authors showed that the complex cohomology groups of the corresponding toric variety as an analytic space coincide with the hypercohomology groups associated to the logarithmic double complex.
Abstract: On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case. Over the field of complex numbers, we prove the toric analog of the algebraic de Rham theorem which Grothendieck formulated and proved for general nonsingular algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely, for a finite simplicial fan which need not be complete, the complex cohomology groups of the corresponding toric variety as an analytic space coincide with the hypercohomology groups of the single complex associated to the logarithmic double complex. They can then be described combinatorially as Ishida's cohomology groups for the fan. We also prove vanishing theorems for Ishida's cohomology groups. As a consequence, we deduce directly that the complex cohomology groups vanish in odd degrees for toric varieties which correspond to finite simplicial fans with full-dimensional convex support. In the particular case of complete simplicial fans, we thus have a direct proof for an earlier result of Danilov and the author.
10 citations
01 Mar 1992
TL;DR: In this paper, the authors classified the weights of non-degenerate quasi-homogeneous polynomials on C which define simple K3 singularities, where the point MO was regarded as the weight of the convex hull.
Abstract: Yonemura [9] classified the weights of non-degenerate quasi-homogeneous polynomials on C which define simple K3 singularities. On the other hand, to each quasi-homogeneous polynomial /—2v(E/z>(y)4Cvz v there exists an element "o in (6>o) such that = 1 if cv^0, where z(>"^TM^) = zF^zTtf Then we may regard the point MO as the weight of/. Let A* be the convex hull of {v
2 citations
01 Mar 1992
01 Jan 1992
TL;DR: Stembridge et al. as mentioned in this paper used marked sequences to encode permutations of the Betti numbers of the cohomology of X(Z) and X(X) of a toric variety.
Abstract: Stembridge, J.R., Eulerian numbers, tableaux, and the Betti numbers of a toric variety, Discrete Mathematics 99 (1992) 307-320. Let Z denote the Coxeter complex of S,, and let X(X) denote the associated toric variety. Since the Betti numbers of the cohomology of X(Z) are Eulerian numbers, the additional presence of an &-module structure permits the definition of an isotypic refinement of these numbers. In some unpublished work, DeConcini and Procesi derived a recurrence for the &-character of the cohomology of X(Z), and Stanley later used this to translate the problem of combinatorially describing the isotypic Eulerian numbers into the language of symmetric functions. In this paper, we explicitly solve this problem by developing a new way to use marked sequences to encode permutations. This encoding also provides a transparent explanation of the unimodality of Eulerian numbers and their isotypic refinements.