Open AccessBook
Birational Geometry of Algebraic Varieties
János Kollár,Shigefumi Mori +1 more
TLDR
In this paper, the authors introduce the minimal model program and the canonical class of rational curves, and present the singularities of the model program, as well as three dimensional flops.Abstract:
1. Rational curves and the canonical class 2. Introduction to minimal model program 3. Cone theorems 4. Surface singularities 5. Singularities of the minimal model program 6. Three dimensional flops 7. Semi-stable minimal models.read more
Citations
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Journal ArticleDOI
Existence of minimal models for varieties of log general type
TL;DR: In this paper, it was shown that pl-flips exist in dimension n − 1, assuming finite generation in dimension N − 1 and assuming that pl flips exist in all dimensions.
Journal Article
Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory
TL;DR: In this article, the authors theoretically compare the Bayes cross-validation loss and the widely applicable information criterion and prove two theorems: 1) The Bayes generalization error is asymptotically equal to 2λ/n, where λ is the real log canonical threshold and n is the number of training samples.
Book
Classical Algebraic Geometry: A Modern View
TL;DR: In this paper, the authors present a survey of the geometry of lines and cubic surfaces, including determinantal equations, theta characteristics, and the Cremona transformations.
Journal ArticleDOI
Monge-Ampère equations in big cohomology classes
TL;DR: In this paper, the authors define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kahler manifold X and show that the solution has minimal singularities in the sense of Demailly if μ has L 1+e-density with respect to Lebesgue measure.
Book
Analytic Methods in Algebraic Geometry
TL;DR: In this paper, the authors describe analytic techniques useful in the study of questions pertaining to linear series, multiplier ideals, and vanishing theorems for algebraic vector bundles, assuming that the reader is already somewhat acquainted with the basic concepts of sheaf theory, homological algebra, and complex differential geometry.