Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

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Existence of minimal models for varieties of log general type

In regular statistical models, the leave-one-out cross-validation is asymptotically equivalent to the Akaike information criterion. However, since many learning machines are singular statistical models, the asymptotic behavior of the cross-validation remains unknown. In previous studies, we established the singular learning theory and proposed a widely applicable information criterion, the expectation value of which is asymptotically equal to the average Bayes generalization loss. In the present paper, we theoretically compare the Bayes cross-validation loss and the widely applicable information criterion and prove two theorems. First, the Bayes cross-validation loss is asymptotically equivalent to the widely applicable information criterion as a random variable. Therefore, model selection and hyperparameter optimization using these two values are asymptotically equivalent. Second, the sum of the Bayes generalization error and the Bayes cross-validation error is asymptotically equal to 2λ/n, where λ is the real log canonical threshold and n is the number of training samples. Therefore the relation between the cross-validation error and the generalization error is determined by the algebraic geometrical structure of a learning machine. We also clarify that the deviance information criteria are different from the Bayes cross-validation and the widely applicable information criterion.

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Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory

/pdf/introduction-to-the-minimal-model-problem-2rzka1dyt7.pdf

Introduction to the Minimal Model Problem

We sketch in this article a new theory, which we call Symplectic Field Theory or SFT, which provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field theory, and at the same time serves as a rich source of new invariants of contact manifolds and their Legendrian submanifolds. Moreover, we hope that the applications of SFT go far beyond this framework.1

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Introduction to Symplectic Field Theory

Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.

Classical Algebraic Geometry: A Modern View

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.

Birational Geometry of Algebraic Varieties

Let X be an arbitrary nonsingular projective 3-fold whose canonical bundle is not numerically effective. Then we have: (i) X contains an exceptional divisor of several types, which we classify explicitly, (ii) X has a morphism to a projective nonsingular surface whose fibers are conics, (iii) X has a morphism to a projective nonsingular curve whose general fibers are Del Pezzo surfaces, or (iv) X is a Fano 3-fold with Picard number 1.

Threefolds whose canonical bundles are not numerically effective

We show that if a flat group scheme acts properly, with finite stabilizers, on an algebraic space, then a quotient exists as a separated algebraic space. More generally we show any flat groupid for which the family of stabilizers is finite has a uniform geometric, uniform categorical quotient in the category of algebraic spaces. Our argument is elementary and essentially self contained.

Quotients by groupoids

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Rational connectedness and boundedness of Fano manifolds

§ O. Introduction § I. Preliminaries and basic definitions § la (Appendix la). Results on 3-fold terminal singularities § Ib (Appendix Ib). Deformation of extremal nbds § 2. Numerical invariants ip(n), wp(O), and w;(n) § 3. Embedding dimension of (en, pn) § 4. Classification of X ::> e at P into cases § 5. Numerical calculations for (IA), (IC), (IA v) , and (IC v) § 6. Possible singularities on an extremal nbd X::> e === Wi § 7. Existence of "good" anti(bi)canonical divisor (easy case) § 8. J -filtration and lifting structure § 9. Existence of "good" anticanonical divisor (delicate case) §IO. A slight generalization of (0.4.5) and comments on (la.7) Appendix A. Summary of local classification of X::> e 3 P Appendix B. Possible singularities on an extremal nbd List of notation and terminology References

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Shigefumi Mori

Papers

Birational Geometry of Algebraic Varieties

Threefolds whose canonical bundles are not numerically effective

Quotients by groupoids

Rational connectedness and boundedness of Fano manifolds

Flip theorem and the existence of minimal models for 3-folds