Hodge Theory and Algebraic Geometry
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Resolving 3-Dimensional Toric Singularities
Resolving 3-dimensional toric singularities ∗Dimitrios I. Dais Mathematics Department, Section of Algebra and Geometry, University of Ioannina GR-45110 Ioannina, Greece. E-mail: [email protected] Abstract This paper surveys, in the first place, some basic facts from the classification theory of normal complex singularities, including details for the low dimensions 2 and 3. Next, it describes how the toric singularities are located within the class of rational singularities, and recalls their main properties. Finally, it focuses, in particular, on a toric version of Reid’s desingularization strategy in dimension three. 1 Introduction There are certain general qualitative criteria available for the rough classification of singularities of complex varieties. The main ones arise: • from the study of the punctual algebraic behaviour of these varieties (w.r.t. local rings associated to singular points) [algebraic classification] • from an intrinsic characterization for the nature of the possible exceptional loci w.r.t. any desingularization [rational, elliptic, non-elliptic etc.] arXiv:math/0110278v1 [math.AG] 25 Oct 2001 • from the behaviour of “discrepancies” (for Q-Gorenstein normal complex varieties) [adjunction-theoretic classification] ∗GSRT-fellow supported by the European Union and the Greek Ministry of Research and Technology. 2000 Mathematics Subject Classification. Primary 14M25; Secondary 14B05, 32S05. Keywords: canonical singularities, toric singularities. 1 2 dimitrios i. dais • Algebraic Classification. At first we recall some fundamental definitions from commutative algebra (cf. [52], [54]). Let R be a commutative ring with 1. The height ht(p) of a prime ideal p of R is the supremum of the lengths of all prime ideal chains which are contained in p, and the dimension of R is defined to be dim (R) := sup {ht (p) |p prime ideal of R } . -
Arxiv:2108.13052V1 [Math.AG]
EFFECTIVE RESULTS IN THE THREE-DIMENSIONAL MINIMAL MODEL PROGRAM YURI PROKHOROV Abstract. We give a brief review on recent developments in the three-dimensional minimal model program. In this note we give a brief review on recent developments in the three-dimensional minimal model program (MMP for short). Certainly, this is not a complete survey of all advances in this area. For example, we do not discuss the minimal models of varieties of non-negative Kodaira dimension, as well as, applications to birational geometry and moduli spaces. The aim of the MMP is to find a good representative in a fixed birational equivalence class of algebraic varieties. Starting with an arbitrary smooth projective variety one can perform a finite number of certain elementary transformations, called divisorial contractions and flips, and at the end obtain a variety which is simpler in some sense. Most parts of the MMP are completed in arbitrary dimension. One of the basic remaining problems is the following: Describe all the intermediate steps and the outcome of the MMP. The MMP makes sense only in dimensions 2 and for surfaces it is classical and well-known. So the first non-trivial case is the three-dimensional≥ one. It turns out that to proceed with the MMP in dimension 3 one has to work with varieties admitting certain types of very mild, so- called terminal, singularities.≥ On the other hand, dimension 3 is the last dimension where one can expect effective results: in higher dimensions classification results become very complicated and unreasonably long. We will work over the field C of complex numbers throughout. -
Birational Geometry
BIRATIONAL GEOMETRY CAUCHER BIRKAR Contents 1. Introduction: overview 2 2. Preliminaries 7 3. Singularities 10 4. Minimal model program 15 4.1. Kodaira dimension 15 4.2. Basics of minimal model program 16 4.3. Cone and contraction, vanishing, nonvanishing, and base point freeness 18 5. Further studies of minimal model program 26 5.1. Finite generation, flips and log canonical models 26 5.2. Termination of flips 29 5.3. Minimal models 31 5.4. Mori fibre spaces 32 6. Appendix 35 References 35 3 April 2007. These are lecture notes for the Birational Geometry course which I taught at the University of Cambridge, Winter 2007. 1 2 CAUCHER BIRKAR 1. INTRODUCTION: OVERVIEW All varieties in this lecture are assumed to be algebraic over C. What classification means? In every branch of mathematics, the problem of classifying the objects arises naturally as the ultimate un- derstanding of the subject. If we have finitely many objects, then clas- sification means trying to see which object is isomorphic to another, whatever the isomorphism means. In this case, the problem shouldn’t be too difficult. However, if we have infinitely many objects, then clas- sification has a bit different meaning. Since each human can live for a finite length of time, we may simply not have enough time to check every object in the theory. On the other hand, objects in a mathe- matical theory are closely related, so one can hope to describe certain properties of all the objects using only finitely many or some of the objects which are nice in some sense. -
Singularities of (L)MMP and Applications Massimiliano Mella
Singularities of (L)MMP and applications Massimiliano Mella Dipartimento di Matematica, Universita` di Ferrara, Italia E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary 14E30; Secondary 14J17, 14B05,14J30 Key words and phrases. terminal, canonical, log pairs, minimal model program Partially supported by “Geometria sulle Variet`aAlgebriche”(MUR). Introduction These lectures are devoted to an introduction to singularities occurring in the various flavors of Minimal Model Program. In these days an enormous interest came back to the program thank to the amazing results of Birkar, Cascini, Hacon, and Mc Kernan, [BCHM]. The Grenoble school is therefore a lucky coincidence to attract even more interest on the subject. The first chapter is a tribute to [YPG]. The wonderful paper of Miles Reid that allowed various generation of mathematician to understand singularities and how to work with them. It is not by chance that after more than twenty years this is still a milestone of the theory. Here I revise the main constructions and results of terminal and canonical 3-folds. My main aim is to give examples and geometric intuition to this technical part of MMP theory. For this reason the first and the last section are, hopefully, half way between examples and exercises one should work out by himself on his desk. The second chapter is dedicated to singularities of pairs. Here it is difficult to put technicalities apart. I hope that the applications to birational and biregular geometry given in the last two sections will help swallow all the definitions. For this part there are various references, the ones that I mainly follow are, [U2], [KM2], and [C+]. -
Balanced Line Bundles on Fano Varieties
BALANCED LINE BUNDLES ON FANO VARIETIES BRIAN LEHMANN, SHO TANIMOTO, AND YURI TSCHINKEL Abstract. A conjecture of Batyrev and Manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants. We analyze the geometry underlying these invariants using the Minimal Model Program and then apply our results to primitive Fano threefolds. 1. Introduction Let X be a smooth projective variety defined over a number field F and L = (L; k · k) an ample, adelically metrized, line bundle on X. Such line bundles give rise to height functions X(F ) ! R>0 x 7! HL(x) on the set of F -rational points (see, e.g., [CLT01, Section 1.3] for the definitions). A basic result is that the associated counting function N(X(F ); L;B) := #fx 2 X(F ) j HL(x) ≤ Bg is finite, for each B 2 R. Conjectures of Manin and Batyrev-Manin concern the asymp- totic behavior of N(X; L;B), as B ! 1, for Fano varieties, i.e., varieties X with ample anticanonical class −KX . The conjectures predict that this asymptotic is controlled by the geometry of X and L [BM90]. More precisely, define the geometric constants a(X; L) = minft 2 R j t[L] + [KX ] 2 Λeff (X)g: and b(X; L) = the codimension of the minimal supported face of Λeff (X) containing the numerical class a(X; L)[L] + [KX ]: The following extension of Manin's original conjecture builds on [BM90], [Pey95], [BT98]: Manin's Conjecture. Let X be a smooth projective variety over a number field, with ample anticanonical class −KX . -
Interaction Between Singularity Theory and the Minimal Model Program
P. I. C. M. – 2018 Rio de Janeiro, Vol. 2 (825–848) INTERACTION BETWEEN SINGULARITY THEORY AND THE MINIMAL MODEL PROGRAM C X (许晨阳) Abstract We survey some recent topics on singularities, with a focus on their connection to the minimal model program. This includes the construction and properties of dual complexes, the proof of the ACC conjecture for log canonical thresholds and the re- cent progress on the ‘local stability theory’ of an arbitrary Kawamata log terminal singularity. 1 Introduction Through out this paper, we will consider algebraic singularities in characteristic 0. It is well known that even if we are mostly interested in smooth varieties, for many different reasons, we have to deal with singular varieties. For the minimal model program (MMP) (also known as Mori’s program), the reason is straightforward, mildly singular varieties are built into the MMP process, and there is no good way to avoid them (see e.g. Kollár and Mori [1998]). In fact, the development of the MMP has been intertwined with the progress of our understanding of the corresponding singularity theory, in particular for the classes of singularities preserved by an MMP sequence. This is one of the main reasons why when the theory was started around four decades ago, people spent a lot of time to classify these singularities. However, once we move beyond dimension three, an explicit characterisation of these singularities is often too complicated, and we have to search for a more intrinsic and qualitative method. It turns out that MMP theory itself provides many new tools for the study of singularities. -
LECTURES on BIRATIONAL GEOMETRY Contents 1. Introduction
LECTURES ON BIRATIONAL GEOMETRY CAUCHER BIRKAR Contents 1. Introduction: overview 3 2. Preliminaries 11 3. Contractions and extremal rays 13 4. Pairs and their singularities 19 5. Kodaira dimension 25 6. The log minimal model program: main conjectures 30 7. Cone and contraction, vanishing, nonvanishing, and base point freeness 37 8. D-flips and local finite generation 49 9. Pl flips and extension theorems 55 10. Existence of log minimal models and Mori fibre spaces 60 11. From nonvanishing to log minimal models 64 12. Finiteness of log minimal models 68 13. Termination with scaling 72 14. The nonvanishing 76 15. A few consequences of Theorem 10.1 81 References 85 Winter 2011. These are lecture notes for a course on birational geometry that I am currently teaching at College de France, Paris. This document will be updated as we progress. Info: University of Cambridge, and Fondation Sciences Math´ematiquesde Paris. Email: [email protected]. 1 2 CAUCHER BIRKAR 1. Introduction: overview All varieties in this lecture are assumed to be algebraic over some algebraically closed field k (and this is going to be C for much of the course). What classification means? In every branch of mathematics, the problem of classifying the objects arises naturally as the ultimate un- derstanding of the subject. If we have finitely many objects, then clas- sification means trying to see which object is isomorphic to another, whatever the isomorphism means. In this case, the problem shouldn't be too difficult. However, if we have infinitely many objects, then clas- sification has a bit different meaning. -
3-Fold Log Flips
Home Search Collections Journals About Contact us My IOPscience 3-FOLD LOG FLIPS This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1993 Russian Acad. Sci. Izv. Math. 40 95 (http://iopscience.iop.org/1468-4810/40/1/A04) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 195.37.209.182 The article was downloaded on 19/09/2010 at 20:19 Please note that terms and conditions apply. H3B Pocc. AKa«. HayK Russian Aead. Sci. Izv. Math. Cep. MaxeM. TOM 56 (1992), J\fc 1 Vol. 40 (1993), No. 1 3-FOLD LOG FLIPS* UDC 512.7 V. V. SHOKUROV ABSTRACT. We prove that 3-fold log flips exist. We deduce the existence of log canon- ical and Q-factorial log terminal models, as well as a positive answer to the inversion problem for log canonical and log terminal adjunction. CONTENTS §0. § 1. Singularities and models §2. The covering trick §3. Adjunction of log divisors §4. Two terminations §5. Complementary log divisors §6. Special flips §7. Exceptional special flips §8. Index 2 special flips §9. Applications Appendix by Y. Kawamata: The minimal discrepancy of a 3-fold terminal singularity §10. Commentary by M. Reid Bibliography §0. Let X be a normal algebraic (or analytic) 3-fold with a marked Q-divisor Β = Βχ (the boundary of X); we write Κ — Κ χ , and consider the log canonical divisor K+B as in Kawamata-Matsuda-Matsuki [8]. Suppose that /: X —* Ζ is a birational contraction of X such that Κ + Β is numerically nonpositive relative to /.