Contents 1. Preliminaries on the Cones of Ample and Effective Divisors 2 2. Preliminaries on the Moduli Space of Curves And
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Introduction
SEMINAR WS 16/17 BERKOVICH SPACES, BIRATIONAL GEOMETRY AND MOTIVIC ZETA FUNCTIONS Talk II on Birational geometry and MMP Efstathia Katsigianni 10.11.2016 Introduction The goal of this talk is to present the notions that have to do with the Minimal model Program and will be used in the course of the seminar, as well as the main theorems of this area. For this we rely entirely on the references listed in the end. Given a projective variety X defined over an algebraically closed field k, one could try to find another projective variety birationally isomorphic to X which is the simplest possible. We define on the set CX of all (isomorphism classes of) projective varieties birationally isomorphic to X a relation as fol- lows: if X0, X1 are in CX , we write X0 < X1, whenever there is a birational morphism X0 ! X1. This defines an ordering on CX and we look for minimal elements in CX or for the smallest element of it. For example, each irreducible curve is birational to a unique smooth pro- jective curve, thus the investigation of smooth projective curves is equivalent to the study of all curves up to birational equivalence. When X is a smooth surface, it has a smooth minimal model obtained by contracting all exceptional curves on it. If X is not uniruled, this minimal model has nef canonical divisor and is the smallest element in cX. When X is uniruled, this minimal model is not unique. Smooth projective varieties with nef canonical bundles are minimal in the above sense: Proposition. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Cox Rings and Partial Amplitude Permalink https://escholarship.org/uc/item/7bs989g2 Author Brown, Morgan Veljko Publication Date 2012 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Cox Rings and Partial Amplitude by Morgan Veljko Brown A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, BERKELEY Committee in charge: Professor David Eisenbud, Chair Professor Martin Olsson Professor Alistair Sinclair Spring 2012 Cox Rings and Partial Amplitude Copyright 2012 by Morgan Veljko Brown 1 Abstract Cox Rings and Partial Amplitude by Morgan Veljko Brown Doctor of Philosophy in Mathematics University of California, BERKELEY Professor David Eisenbud, Chair In algebraic geometry, we often study algebraic varieties by looking at their codimension one subvarieties, or divisors. In this thesis we explore the relationship between the global geometry of a variety X over C and the algebraic, geometric, and cohomological properties of divisors on X. Chapter 1 provides background for the results proved later in this thesis. There we give an introduction to divisors and their role in modern birational geometry, culminating in a brief overview of the minimal model program. In chapter 2 we explore criteria for Totaro's notion of q-amplitude. A line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m0 such that m ≥ m0 implies Hi(X; F ⊗ O(mL)) = 0 for i > q. -
Effective Bounds for the Number of MMP-Series of a Smooth Threefold
EFFECTIVE BOUNDS FOR THE NUMBER OF MMP-SERIES OF A SMOOTH THREEFOLD DILETTA MARTINELLI Abstract. We prove that the number of MMP-series of a smooth projective threefold of positive Kodaira dimension and of Picard number equal to three is at most two. 1. Introduction Establishing the existence of minimal models is one of the first steps towards the birational classification of smooth projective varieties. More- over, starting from dimension three, minimal models are known to be non-unique, leading to some natural questions such as: does a variety admit a finite number of minimal models? And if yes, can we fix some parameters to bound this number? We have to be specific in defining what we mean with minimal mod- els and how we count their number. A marked minimal model of a variety X is a pair (Y,φ), where φ: X 99K Y is a birational map and Y is a minimal model. The number of marked minimal models, up to isomorphism, for a variety of general type is finite [BCHM10, KM87]. When X is not of general type, this is no longer true, [Rei83, Example 6.8]. It is, however, conjectured that the number of minimal models up to isomorphism is always finite and the conjecture is known in the case of threefolds of positive Kodaira dimension [Kaw97]. In [MST16] it is proved that it is possible to bound the number of marked minimal models of a smooth variety of general type in terms of the canonical volume. Moreover, in dimension three Cascini and Tasin [CT18] bounded the volume using the Betti numbers. -
AN INTRODUCTION to MORI THEORY: the CASE of SURFACES Contents Introduction 1 Part 1. Preliminaries 2 1.1. Basic Notation 3 1.2
AN INTRODUCTION TO MORI THEORY: THE CASE OF SURFACES MARCO ANDREATTA Contents Introduction 1 Part 1. Preliminaries 2 1.1. Basic notation 3 1.2. Morphisms associated to line bundles; ample line bundles 4 1.3. Riemann - Roch and Hodge Index theorems 5 Part 2. Mori theory for surfaces 6 2.1. Mori-Kleiman cone 6 2.2. Examples 9 2.3. The rationality lemma and the cone theorem 11 2.4. Castelnuovo contraction theorem 16 2.5. Base point freeness, BPF 16 2.6. Minimal model program for surfaces 20 Part 3. Birational theory for surfaces 22 3.1. Castelnuovo rationality criterium 22 3.2. Factorization of birational morphisms 23 3.3. Singularities and log singularities. 25 3.4. Castelnuovo-Noether theorem and some definitions in the Sarkisov program 26 References 32 Introduction One of the main results of last decades algebraic geometry was the foun- dation of Mori theory or Minimal Model Program, for short MMP, and its proof in dimension three. Minimal Model Theory shed a new light on what 1991 Mathematics Subject Classification. Primary 14E30, 14J40 ; Secondary 14J30, 32H02. Key words and phrases. Fano Mori spaces, contractions, base point freeness, sub- adjunction, extremal ray. 1 2 MARCO ANDREATTA is nowadays called higher dimensional geometry. In mathematics high num- bers are really a matter of circumstances and here we mean greater than or equal to 3. The impact of MMP has been felt in almost all areas of algebraic geometry. In particular the philosophy and some of the main new objects like extremal rays, Fano-Mori contractions or spaces and log varieties started to play around and give fruitful answer to different problems. -
Resolutions of Quotient Singularities and Their Cox Rings Extended Abstract
RESOLUTIONS OF QUOTIENT SINGULARITIES AND THEIR COX RINGS EXTENDED ABSTRACT MAKSYMILIAN GRAB The resolution of singularities allows one to modify an algebraic variety to obtain a nonsingular variety sharing many properties with the original, singular one. The study of nonsingular varieties is an easier task, due to their good geometric and topological properties. By the theorem of Hironaka [29] it is always possible to resolve singularities of a complex algebraic variety. The dissertation is concerned with resolutions of quotient singularities over the field C of complex numbers. Quotient singularities are the singularities occuring in quotients of algebraic varieties by finite group actions. A good local model of such a singularity n is a quotient C /G of an affine space by a linear action of a finite group. The study of quotient singularities dates back to the work of Hilbert and Noether [28], [40], who proved finite generation of the ring of invariants of a linear finite group action on a ring of polynomials. Other important classical result is due to Chevalley, Shephard and Todd [10], [46], who characterized groups G that yield non-singular quotients. Nowa- days, quotient singularities form an important class of singularities, since they are in some sense typical well-behaved (log terminal) singularities in dimensions two and three (see [35, Sect. 3.2]). Methods of construction of algebraic varieties via quotients, such as the classical Kummer construction, are being developed and used in various contexts, including higher dimensions, see [1] and [20] for a recent example of application. In the well-studied case of surfaces every quotient singularity admits a unique minimal resolution. -
Algebraic Surfaces and Hyperbolic Geometry
Current Developments in Algebraic Geometry MSRI Publications Volume 59, 2011 Algebraic surfaces and hyperbolic geometry BURT TOTARO We describe the Kawamata–Morrison cone conjecture on the structure of Calabi–Yau varieties and more generally klt Calabi–Yau pairs. The conjecture is true in dimension 2. We also show that the automorphism group of a K3 surface need not be commensurable with an arithmetic group, which answers a question by Mazur. 1. Introduction Many properties of a projective algebraic variety can be encoded by convex cones, such as the ample cone and the cone of curves. This is especially useful when these cones have only finitely many edges, as happens for Fano varieties. For a broader class of varieties which includes Calabi–Yau varieties and many rationally connected varieties, the Kawamata–Morrison cone conjecture predicts the structure of these cones. I like to think of this conjecture as what comes after the abundance conjecture. Roughly speaking, the cone theorem of Mori, Kawamata, Shokurov, Kollár, and Reid describes the structure of the curves on a projective variety X on which the canonical bundle K X has negative degree; the abundance conjecture would give strong information about the curves on which K X has degree zero; and the cone conjecture fully describes the structure of the curves on which K X has degree zero. We give a gentle summary of the proof of the cone conjecture for algebraic surfaces, with plenty of examples [Totaro 2010]. For algebraic surfaces, these cones are naturally described using hyperbolic geometry, and the proof can also be formulated in those terms. -
The Birational Geometry of Tropical Compactifications
University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations Spring 2010 The Birational Geometry of Tropical Compactifications Colin Diemer University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Algebraic Geometry Commons Recommended Citation Diemer, Colin, "The Birational Geometry of Tropical Compactifications" (2010). Publicly Accessible Penn Dissertations. 96. https://repository.upenn.edu/edissertations/96 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/96 For more information, please contact [email protected]. The Birational Geometry of Tropical Compactifications Abstract We study compactifications of subvarieties of algebraic tori using methods from the still developing subject of tropical geometry. Associated to each ``tropical" compactification is a polyhedral object called a tropical fan. Techniques developed by Hacking, Keel, and Tevelev relate the polyhedral geometry of the tropical variety to the algebraic geometry of the compactification. eW compare these constructions to similar classical constructions. The main results of this thesis involve the application of methods from logarithmic geometry in the sense of Iitaka \cite{iitaka} to these compactifications. eW derive a precise formula for the log Kodaira dimension and log irregularity in terms of polyhedral geometry. We then develop a geometrically motivated theory of tropical morphisms and discuss the induced map on tropical fans. Tropical fans with similar structure in this sense are studied, and we show that certain natural operations on a tropical fan correspond to log flops in the sense of birational geometry. These log flops are then studied via the theory of secondary polytopes developed by Gelfand, Kapranov, and Zelevinsky to obtain polyhedral analogues of some results from logarithmic Mori theory. -
Endomorphisms of Smooth Projective 3-Folds with Non-Negative Kodaira Dimension
i i Publ. RIMS, Kyoto Univ. 38 (2002), 33–92 Endomorphisms of Smooth Projective 3-Folds with Non-Negative Kodaira Dimension By Yoshio Fujimoto∗ Abstract Let X be a nonsingular projective 3-fold with non-negative Kodaira dimension κ(X) ≥ 0 which admits a nonisomorphic surjective morphism f : X → X onto itself. If κ(X) = 0 or 2, a suitable finite ´etale covering X˜ of X is isomorphic to an abelian 3-fold or the direct product E × S of an elliptic curve E and a nonsingular algebraic surface S with κ(S)=κ(X). §1. Introduction The main purpose of this paper is to study the structure of a nonsingular projective 3-fold X with a surjective morphism f : X → X onto itself which is not an isomorphism. We call it a nontrivial surjective endomorphism of X. Let f : X → X be a surjective morphism from a nonsingular projective variety X onto itself. Then f is a finite morphism and if the Kodaira dimension κ(X)ofX is non-negative, f is a finite ´etale covering. Moreover, if X is of general type or a compact hyperbolic manifold in the sense of Kobayashi [Kob], then X admits no nontrivial surjective endomorphisms (cf. [I], [Kob]). Surjec- tive endomorphisms of a projective manifold with negative Kodaira dimension have been studied by many algebraic geometers (cf. [CS], [D], [HM], [PS]) from the viewpoint of Lazarsfeld’s conjecture (cf. [PS], Problem 1). Notably, in [HM, Main Theorem] and [PS, Proposition 2], it has been proved that a ra- tional homogeneous manifold of Picard number 1 has no nontrivial surjective endomorphisms unless it is isomorphic to the complex projective space Pn. -
Arxiv:1503.08320V2 [Math.AG] 10 Dec 2015 O Hsnto;W Eegie Ytecnet Flrecmlxstr Complex Large of Symmetry
December 11, 2015 THE DUAL COMPLEX OF CALABI–YAU PAIRS JANOS´ KOLLAR´ AND CHENYANG XU Abstract. A log Calabi–Yau pair consists of a proper variety X and a divisor D on it such that KX +D is numerically trivial. A folklore conjecture predicts that the dual complex of D is homeomorphic to the quotient of a sphere by a finite group. The main result of the paper shows that the fundamental group of the dual complex of D is a quotient of the fundamental group of the smooth locus of X, hence its pro-finite completion is finite. This leads to a positive answer in dimension ≤ 4. We also study the dual complex of degenerations of Calabi–Yau varieties. The key technical result we prove is that, after a volume preserving birational equivalence, the transform of D supports an ample divisor. A log Calabi–Yau pair, abbreviated as logCY, is a pair (X, ∆) consisting of a proper variety X and an effective Q-divisor ∆ such that (X, ∆) is log canonical and KX + ∆ is Q-linearly equivalent to 0. Any Calabi–Yau variety X can be naturally identified with the log Calabi–Yau pair (X, 0). At the other extreme, if X is a Fano variety and ∆ ∼Q −KX is an effective divisor then (X, ∆) is also logCY (provided that it is log canonical). Here we are interested in these Fano–type logCYs, especially when ∆ is “large.” Being Fano is not preserved under birational equivalence, thus it is better to define “large” without reference to Fano varieties. There are several natural candidates for this notion; we were guided by the concepts of large complex structure limit and maximal unipotent degeneration used in Mirror Symmetry. -
ON the EXISTENCE of FLIPS 1. Introduction the Main Result of This Paper Is: Theorem 1.1. Assume the Real MMP in Dimension
ON THE EXISTENCE OF FLIPS CHRISTOPHER D. HACON AND JAMES MCKERNAN Abstract. Using the techniques of [17], [7], [13] and [15], we prove that flips exist in dimension n, if one assumes the termination of real flips in dimension n − 1. 1. Introduction The main result of this paper is: Theorem 1.1. Assume the real MMP in dimension n − 1. Then flips exist in dimension n. Here are two consequences of this result: Corollary 1.2. Assume termination of real flips in dimension n − 1 and termination of flips in dimension n. Then the MMP exists in dimension n. As Shokurov has proved, [14], the termination of real flips in di- mension three, we get a new proof of the following result of Shokurov [15]: Corollary 1.3. Flips exist in dimension four. Given a proper variety, it is natural to search for a good birational model. An obvious, albeit hard, first step is to pick a smooth projective model. Unfortunately there are far too many such models; indeed given any such, we can construct infinitely many more, simply by virtue of successively blowing up smooth subvarieties. To construct a unique model, or at least cut down the examples to a manageable number, we have to impose some sort of minimality on the birational model. The choice of such a model depends on the global geometry of X. One possibility is that we can find a model on which the canonical di- visor KX is nef, so that its intersection with any curve is non-negative. Date: December 31, 2005. -
SINGULARITIES of PAIRS János Kollár Contents 1. Introduction 2
SINGULARITIES OF PAIRS Janos´ Kollar´ Contents 1. Introduction 2. Vanishing Theorems 3. Singularities of Pairs 4. Bertini Theorems 5. Effective Base Point Freeness 6. Construction of Singular Divisors 7. The L2 Extension Theorem and Inversion of Adjunction 8. The Log Canonical Threshold 9. The Log Canonical Threshold and the Complex Singular Index 10. The Log Canonical Threshold and the Bernstein-Sato Polynomial 11. Rational and Canonical Singularities 1. Introduction Higher dimensional algebraic geometry has been one of the most rapidly devel- oping research areas in the past twenty years. The first decade of its development centered around the formulation of the minimal model program and finding tech- niques to carry this program through. The proof of the existence of flips, given in [Mori88], completed the program in dimension three. These results, especially the progress leading up to [Mori88], are reviewed in several surveys. A very gen- eral overview is given in [Koll´ar87b];many of the methods are explained in the series of lectures [CKM88]; a technically complete review for experts is found in [KaMaMa87]. The methods of [Mori88] allow us to understand three dimensional flips, but the question of how to proceed to higher dimensions remains a baffling one. Therefore the focus of the field moved in one of two major directions. Internal Developments. There has been a considerable internal development, as we have understood the implications of the minimal model program to the structure of threefolds. Two major achievements in this direction are the study of log flips by [Shokurov92], and its subsequent application to the proof of the abundance theorem for threefolds in [Kawamata92]. -
Pseudoeffective Cones and Morphisms of Projective Varieties
Pseudoeffective Cones and Morphisms of Projective Varieties by Ashwath Rabindranath A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in the University of Michigan 2018 Doctoral Committee: Professor Mircea Mustata, Chair Associate Professor Bhargav Bhatt Professor Mattias Jonsson Professor Karen Smith Associate Professor James Tappenden Ashwath Rabindranath [email protected] ORCID id: 0000-0002-4431-5720 © Ashwath Rabindranath 2018 ACKNOWLEDGMENTS First, I want to thank my advisor, Mircea Mustata who has served as my teacher, my mentor, and as a huge influence in my career. He has guided me through good times and bad and has helped me survive the darkest hours of writing this thesis. I especially want to thank him for reading the same introduction numerous times until I got it right. Moreover, his insight and invaluable advice have greatly influenced me and made me a better mathematician. I would like to thank all of the faculty at Michigan and elsewhere who have been my teachers and mentors, including but not limited to Bhargav Bhatt, Manjul Bhargava, Bill Fulton, Mel Hochster, Mattias Jonsson, Yusuf Mustopa, Karen Smith and Shou-Wu Zhang. During graduate school, I have shared great math conversations with my peers: Harold Blum, Jake Levinson, Takumi Murayama, Weichen Gu, Brooke Ullery, Felipe Perez, Emanuel Reinecke, David Stapleton, Matt Stevenson, Phil Tosteson, John Wiltshire- Gordon, Rachel Karpman, Ming Zhang and many more. Finally, I want to thank my parents and brother for their invaluable moral support during these challenging years. ii TABLE OF CONTENTS Acknowledgments ................................... ii List of Figures ....................................