Algebraic Geometry

Hartshorne, Robin

doi:10.1007/978-1-4757-3849-0

Cited by 7,167 publications

(5,026 citation statements)

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“…In the present paper, we will not pursue this very case-dependent possibility further. 4 This follows directly from the Snake Lemma [23], using the obvious injections of B, C into B ⊕ R and C ⊕ R.…”

Section: Positivitymentioning

confidence: 98%

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Monad bundles in heterotic string compactifications

Anderson

Lukas

2008

J. High Energy Phys.

106

204

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In this paper, we study positive monad vector bundles on complete intersection Calabi-Yau manifolds in the context of E 8 × E 8 heterotic string compactifications. We show that the class of such bundles, subject to the heterotic anomaly condition, is finite and consists of about 7000 models. We explain how to compute the complete particle spectrum for these models. In particular, we prove the absence of vector-like family anti-family pairs in all cases. We also verify a set of highly non-trivial necessary conditions for the stability of the bundles. A full stability proof will appear in a companion paper. A scan over all models shows that even a few rudimentary physical constraints reduces the number of viable models drastically.anderson@maths.ox.ac.uk

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Section: Positivitymentioning

confidence: 98%

“…We invoke the Euler sequence which states that, for an embedding of X into an ambient space A, there is a short exact sequence 23) where N is the normal bundle of X in A and T A is the the tangent bundle of A. The bar and the subscript, X, denotes restriction of the bundle to the Calabi-Yau manifold X.…”

Section: C2 Hodge Numbersmentioning

confidence: 99%

Monad bundles in heterotic string compactifications

Anderson

Lukas

2008

J. High Energy Phys.

106

204

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“…While most of us will know Riemann surfaces from a basic course on complex analysis or algebraic geometry, this will be mainly from an abstract viewpoint like in [86] or [129], respectively. Here we will need a more "computational" approach and I hope that the reader can extract this knowledge from Appendix A.…”

Section: Appendixmentioning

confidence: 99%

Jacobi Operators and Completely Integrable Nonlinear Lattices

Teschl

1999

Mathematical Surveys and Monographs

324

349

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Abstract. This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy.Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m-functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, subordinacy, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations.Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem, solutions in terms of Riemann theta functions, the inverse scattering transform, Bäcklund transformations, and soliton solutions are discussed.Corrections and complements to this book are available from:

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“…1.1]. For n > 0, Proj i∈N R ni = Proj i∈N R i by [16,II Ex. 5.13] (the hypothesis there is not needed for the first statement); similarly for n < 0, Proj i∈N R ni = Proj i∈N R −i .…”

Section: The Simplest Casementioning

confidence: 99%

Geometric invariant theory and flips

Thaddeus

1996

J. Amer. Math. Soc.

288

283

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Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. In light of its fundamental and elementary nature, this is a rather surprising gap, and this paper will attempt to fill it.In one sense, the question can be answered almost completely. Roughly, the space of all possible linearizations is divided into finitely many polyhedral chambers within which the quotient is constant (2.3), (2.4), and when a wall between two chambers is crossed, the quotient undergoes a birational transformation which, under mild conditions, is a flip in the sense of Mori (3.3). Moreover, there are sheaves of ideals on the two quotients whose blow-ups are both isomorphic to a component of the fibred product of the two quotients over the quotient on the wall (3.5). Thus the two quotients are related by a blow-up followed by a blow-down.The ideal sheaves cannot always be described very explicitly, but there is not much more to say in complete generality. To obtain more concrete results, we require smoothness, and certain conditions on the stabilizers which, though fairly strong, still include many interesting examples. The heart of the paper, § §4 and 5, is devoted to describing the birational transformations between quotients as explicitly as possible under these hypotheses. In the best case (4.8), for example, the blow-ups turn out to be just the ordinary blow-ups of certain explicit smooth subvarieties, which themselves have the structure of projective bundles.The last three sections of the paper put this theory into practice, using it to study moduli spaces of points on the line, parabolic bundles on curves, and Bradlow pairs. An important theme is that the structure of each individual quotient is illuminated by understanding the structure of the whole family. So even if there is one especially natural linearization, the problem is still interesting. Indeed, even if the linearization is unique, useful results can be produced by enlarging the variety on which the group acts, so as to create more linearizations.I believe that this problem is essentially elementary in nature, and I have striven to solve it using a minimum of technical machinery. For example, stability and semistability are distinguished as little as possible. Moreover, transcendental methods, choosing a maximal torus, and invoking the numerical criterion are completely avoided. The only technical tool relied on heavily is the marvelous Luna slice theorem [19]. Luckily, this is not too difficult itself, and there is a good exposition in GIT, appendix 1D. This theorem is used, for example, to give a new, easy proof of the Bialynicki-Birula decomposition theorem (1.12).

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Algebraic Geometry

Cited by 7,167 publications

References 0 publications

Monad bundles in heterotic string compactifications

Monad bundles in heterotic string compactifications

Jacobi Operators and Completely Integrable Nonlinear Lattices

Geometric invariant theory and flips

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