CLASSIFICATION of ALGEBRAIC VARIETIES, I1 Kenji Ueno 2
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Recent Results in Higher-Dimensional Birational Geometry
Complex Algebraic Geometry MSRI Publications Volume 28, 1995 Recent Results in Higher-Dimensional Birational Geometry ALESSIO CORTI ct. Abstra This note surveys some recent results on higher-dimensional birational geometry, summarising the views expressed at the conference held at MSRI in November 1992. The topics reviewed include semistable flips, birational theory of Mori fiber spaces, the logarithmic abundance theorem, and effective base point freeness. Contents 1. Introduction 2. Notation, Minimal Models, etc. 3. Semistable Flips 4. Birational theory of Mori fibrations 5. Log abundance 6. Effective base point freeness References 1. Introduction The purpose of this note is to survey some recent results in higher-dimensional birational geometry. A glance to the table of contents may give the reader some idea of the topics that will be treated. I have attempted to give an informal presentation of the main ideas, emphasizing the common grounds, addressing a general audience. In 3, I could not resist discussing some details that perhaps § only the expert will care about, but hopefully will also introduce the non-expert reader to a subtle subject. Perhaps the most significant trend in Mori theory today is the increasing use, more or less explicit, of the logarithmic theory. Let me take this opportunity This work at the Mathematical Sciences Research Institute was supported in part by NSF grant DMS 9022140. 35 36 ALESSIO CORTI to advertise the Utah book [Ko], which contains all the recent software on log minimal models. Our notation is taken from there. I have kept the bibliography to a minimum and made no attempt to give proper credit for many results. -
Introduction to Birational Geometry of Surfaces (Preliminary Version)
INTRODUCTION TO BIRATIONAL GEOMETRY OF SURFACES (PRELIMINARY VERSION) JER´ EMY´ BLANC (Very) quick introduction Let us recall some classical notions of algebraic geometry that we will need. We recommend to the reader to read the first chapter of Hartshorne [Har77] or another book of introduction to algebraic geometry, like [Rei88] or [Sha94]. We fix a ground field k. All results of the first 4 sections work over any alge- braically closed field, and a few also on non-closed fields. In the last section, the case of all perfect fields will be discussed. For our purpose, the characteristic is not important. n n 0.1. Affine varieties. The affine n-space Ak, or simply A , is the set of n-tuples n of elements of k. Any point x 2 A can be written as x = (x1; : : : ; xn), where x1; : : : ; xn 2 k are the coordinates of x. An algebraic set X ⊂ An is the locus of points satisfying a set of polynomial equations: n X = (x1; : : : ; xn) 2 A f1(x1; : : : ; xn) = ··· = fk(x1; : : : ; xn) = 0 where each fi 2 k[x1; : : : ; xn]. We denote by I(X) ⊂ k[x1; : : : ; xn] the set of polynomials vanishing along X, it is an ideal of k[x1; : : : ; xn], generated by the fi. We denote by k[X] or O(X) the set of algebraic functions X ! k, which is equal to k[x1; : : : ; xn]=I(X). An algebraic set X is said to be irreducible if any writing X = Y [ Z where Y; Z are two algebraic sets implies that Y = X or Z = X. -
Threefolds of Kodaira Dimension One Such That a Small Pluricanonical System Do Not Define the Iitaka fibration
THREEFOLDS OF KODAIRA DIMENSION ONE HSIN-KU CHEN Abstract. We prove that for any smooth complex projective threefold of Kodaira di- mension one, the m-th pluricanonical map is birational to the Iitaka fibration for every m ≥ 5868 and divisible by 12. 1. Introduction By the result of Hacon-McKernan [14], Takayama [25] and Tsuji [26], it is known that for any positive integer n there exists an integer rn such that if X is an n-dimensional smooth complex projective variety of general type, then |rKX| defines a birational morphism for all r ≥ rn. It is conjectured in [14] that a similar phenomenon occurs for any projective variety of non-negative Kodaira dimension. That is, for any positive integer n there exists a constant sn such that, if X is an n-dimensional smooth projective variety of non-negative Kodaira dimension and s ≥ sn is sufficiently divisible, then the s-th pluricanonical map of X is birational to the Iitaka fibration. We list some known results related to this problem. In 1986, Kawamata [17] proved that there is an integer m0 such that for any terminal threefold X with Kodaira dimension zero, the m0-th plurigenera of X is non-zero. Later on, Morrison proved that one can 5 3 2 take m0 =2 × 3 × 5 × 7 × 11 × 13 × 17 × 19. See [21] for details. In 2000, Fujino and Mori [12] proved that if X is a smooth projective variety with Kodaira dimension one and F is a general fiber of the Iitaka fibration of X, then there exists a integer M, which depends on the dimension of X, the middle Betti number of some finite covering of F and the smallest integer so that the pluricanonical system of F is non-empty, such that the M-th pluricanonical map of X is birational to the Iitaka fibration. -
Fano Hypersurfaces with Arbitrarily Large Degrees of Irrationality 3
FANO HYPERSURFACES WITH ARBITRARILY LARGE DEGREES OF IRRATIONALITY NATHAN CHEN AND DAVID STAPLETON Introduction There has been a great deal of interest in studying questions of rationality of various flavors. Recall that an n-dimensional variety X is rational if it is birational to Pn and ruled if it is birational to P1 Z for some variety Z . Let X Pn+1 be a degree d hypersurface. × ⊂ In [8], Kollár proved that when X is very general and d 2 3 n + 3 then X is not ruled ≥ ( / )( ) (and thus not rational). Recently these results were generalized and improved by Totaro [14] and subsequently by Schreieder [13]. Schreieder showed that when X is very general and d log n + 2 then X is not even stably rational. In a positive direction, Beheshti and Riedl ≥ 2( ) [2] proved that when X is smooth and n 2d! then X is at least unirational, i.e., dominated ≥ by a rational variety. Given a variety X whose non-rationality is known, one can ask if there is a way to measure “how irrational" X is. One natural invariant in this direction is the degree of irrationality, defined as irr X ≔ min δ > 0 ∃ degree δ rational dominant map X d Pn . ( ) { | } For instance, Bastianelli, De Poi, Ein, Lazarsfeld, and Ullery [1] computed the degree of irrationality for very general hypersurfaces of degree d 2n + 1 by using the positivity of the ≥ canonical bundle. Naturally, one is tempted to ask what can be proved about hypersurfaces with negative canonical bundle. Let X Pn+1 be a smooth hypersurface of degree d. -
MINIMAL MODEL THEORY of NUMERICAL KODAIRA DIMENSION ZERO Contents 1. Introduction 1 2. Preliminaries 3 3. Log Minimal Model Prog
MINIMAL MODEL THEORY OF NUMERICAL KODAIRA DIMENSION ZERO YOSHINORI GONGYO Abstract. We prove the existence of good minimal models of numerical Kodaira dimension 0. Contents 1. Introduction 1 2. Preliminaries 3 3. Log minimal model program with scaling 7 4. Zariski decomposition in the sense of Nakayama 9 5. Existence of minimal model in the case where º = 0 10 6. Abundance theorem in the case where º = 0 12 References 14 1. Introduction Throughout this paper, we work over C, the complex number ¯eld. We will make use of the standard notation and de¯nitions as in [KM] and [KMM]. The minimal model conjecture for smooth varieties is the following: Conjecture 1.1 (Minimal model conjecture). Let X be a smooth pro- jective variety. Then there exists a minimal model or a Mori ¯ber space of X. This conjecture is true in dimension 3 and 4 by Kawamata, Koll¶ar, Mori, Shokurov and Reid (cf. [KMM], [KM] and [Sho2]). In the case where KX is numerically equivalent to some e®ective divisor in dimen- sion 5, this conjecture is proved by Birkar (cf. [Bi1]). When X is of general type or KX is not pseudo-e®ective, Birkar, Cascini, Hacon Date: 2010/9/21, version 4.03. 2000 Mathematics Subject Classi¯cation. 14E30. Key words and phrases. minimal model, the abundance conjecture, numerical Kodaira dimension. 1 2 YOSHINORI GONGYO and McKernan prove Conjecture 1.1 for arbitrary dimension ([BCHM]). Moreover if X has maximal Albanese dimension, Conjecture 1.1 is true by [F2]. In this paper, among other things, we show Conjecture 1.1 in the case where º(KX ) = 0 (for the de¯nition of º, see De¯nition 2.6): Theorem 1.2. -
Arxiv:2008.08852V2 [Math.AG] 15 Oct 2020 Assuming Theorem 2, We Conclude That M16 Cannot Be of General Type, Thus Es- Tablishing Theorem 1
ON THE KODAIRA DIMENSION OF M16 GAVRIL FARKAS AND ALESSANDRO VERRA ABSTRACT. We prove that the moduli space of curves of genus 16 is not of general type. The problem of determining the nature of the moduli space Mg of stable curves of genus g has long been one of the key questions in the field, motivating important developments in moduli theory. Severi [Sev] observed that Mg is unirational for g ≤ 10, see [AC] for a modern presentation. Much later, in the celebrated series of papers [HM], [H], [EH], Harris, Mumford and Eisenbud showed that Mg is of general type for g ≥ 24. Very recently, it has been showed in [FJP] that both M22 and M23 are of general type. On the other hand, due to work of Sernesi [Ser], Chang-Ran [CR1], [CR2] and Verra [Ve] it is known that Mg is unirational also for 11 ≤ g ≤ 14. Finally, Bruno and Verra [BV] proved that M15 is rationally connected. Our result is the following: Theorem 1. The moduli space M16 of stable curves of genus 16 is not of general type. A few comments are in order. The main result of [CR3] claims that M16 is unir- uled. It has been however recently pointed out by Tseng [Ts] that the key calculation in [CR3] contains a fatal error, which genuinely reopens this problem (after 28 years)! Before explaining our strategy of proving Theorem 1, recall the standard notation ∆0;:::; ∆ g for the irreducible boundary divisors on Mg, see [HM]. Here ∆0 denotes b 2 c the closure in Mg of the locus of irreducible 1-nodal curves of arithmetic genus g. -
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. -
Birational Geometry of the Moduli Spaces of Curves with One Marked Point
The Dissertation Committee for David Hay Jensen Certi¯es that this is the approved version of the following dissertation: BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT Committee: Sean Keel, Supervisor Daniel Allcock David Ben-Zvi Brendan Hassett David Helm BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT by David Hay Jensen, B.A. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Ful¯llment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN May 2010 To Mom, Dad, and Mike Acknowledgments First and foremost, I would like to thank my advisor, Sean Keel. His sug- gestions, perspective, and ideas have served as a constant source of support during my years in Texas. I would also like to thank Gavril Farkas, Joe Harris, Brendan Hassett, David Helm and Eric Katz for several helpful conversations. iv BIRATIONAL GEOMETRY OF THE MODULI SPACES OF CURVES WITH ONE MARKED POINT Publication No. David Hay Jensen, Ph.D. The University of Texas at Austin, 2010 Supervisor: Sean Keel We construct several rational maps from M g;1 to other varieties for 3 · g · 6. These can be thought of as pointed analogues of known maps admitted by M g. In particular, they contract pointed versions of the much- studied Brill-Noether divisors. As a consequence, we show that a pointed 1 Brill-Noether divisor generates an extremal ray of the cone NE (M g;1) for these speci¯c values of g. -
The Kodaira Dimension of Siegel Modular Varieties of Genus 3 Or Higher
The Kodaira dimension of Siegel modular varieties of genus 3 or higher Vom Fachbereich Mathematik der Universitat¨ Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von Dipl.-Math. Eric Schellhammer geboren am 18. November 1973 in Stuttgart 2004 [email protected] Universitat¨ Hannover – Fachbereich Mathematik February 25, 2004 Referent: Prof. Dr. Klaus Hulek, Universitat¨ Hannover Koreferent: Prof. Dr. Herbert Lange, Universitat¨ Erlangen Tag der Promotion: 6. Februar 2004 2 Abstract An Abelian variety is a g-dimensional complex torus which is a projective variety. In order to obtain an embedding into projective space one has to chose an ample line bundle. To each such line bundle one can associate a polarisation which only depends on the class of the line bundle in the Neron-Se´ veri group. The type of the polarisation is given by a g-tuple of integers e1;::: ;eg with the property that ei ei+1 for i = 1;:::;g 1. j ei+1 − If e1 = = eg = 1, the polarisation is said to be principal. If the values are ··· ei pairwise coprime, the polarisation is said to be coprime. Furthermore, we can give a symplectic basis for the group of n-torsion points, which is then called a level-n structure. Not only Abelian varieties but also their moduli spaces have been of interest for many years. The moduli space of Abelian varieties with a fixed polarisation (and op- tionally a level-n structure for a fixed n) can be constructed from the Siegel upper half space by dividing out the action of the appropriate arithmetic symplectic group. -
Kodaira Dimension in Low Dimensional Topology 3
KODAIRA DIMENSION IN LOW DIMENSIONAL TOPOLOGY TIAN-JUN LI Abstract. This is a survey on the various notions of Kodaira dimen- sion in low dimensional topology. The focus is on progress after the 2006 survey [78]. Contents 1. Introduction 1 2. κt and κs 2 2.1. The topological Kod dim κt for manifolds up to dimension 3 2 2.2. The symplectic Kod dim κs for 4−manifolds 4 2.3. Symplectic manifolds of dimension 6 and higher 6 3. Calculating κs 6 3.1. Additivity and subadditivity 6 3.2. Behavior under Surgeries 7 4. Main problems and progress in each class 8 4.1. Surfaces and symmetry of κ = −∞ manifolds 8 4.2. Towards the classification of κ = 0 manifolds 11 4.3. Euler number and decomposition of κs = 1 manifolds 15 4.4. Geography and exotic geography of κs = 2 manifolds 16 5. Extensions 17 5.1. Relative Kodaira dim for a symplectic pair 17 5.2. Symplectic manifolds with concave boundary 19 References 19 arXiv:1511.04831v1 [math.SG] 16 Nov 2015 1. Introduction Roughly speaking, a Kodaira dimension type invariant on a class of n−dimensional manifolds is a numerical invariant taking values in the fi- nite set n {−∞, 0, 1, · · · , ⌊ ⌋}, 2 where ⌊x⌋ is the largest integer bounded by x. Date: November 17, 2015. 1 2 TIAN-JUN LI The first invariant of this type is due to Kodaira ([64]) for smooth alge- braic varieties (naturally extended to complex manifolds): Suppose (M, J) is a complex manifold of real dimension 2m. The holomorphic Kodaira dimension κh(M, J) is defined as follows: −∞ if Pl(M, J) = 0 for all l ≥ 1, h κ (M, J)= 0 if Pl(M, J) ∈ {0, 1}, but 6≡ 0 for all l ≥ 1, k k if Pl(M, J) ∼ cl ; c> 0. -
Positivity in Algebraic Geometry I
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 48 Positivity in Algebraic Geometry I Classical Setting: Line Bundles and Linear Series Bearbeitet von R.K. Lazarsfeld 1. Auflage 2004. Buch. xviii, 387 S. Hardcover ISBN 978 3 540 22533 1 Format (B x L): 15,5 x 23,5 cm Gewicht: 1650 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Introduction to Part One Linear series have long stood at the center of algebraic geometry. Systems of divisors were employed classically to study and define invariants of pro- jective varieties, and it was recognized that varieties share many properties with their hyperplane sections. The classical picture was greatly clarified by the revolutionary new ideas that entered the field starting in the 1950s. To begin with, Serre’s great paper [530], along with the work of Kodaira (e.g. [353]), brought into focus the importance of amplitude for line bundles. By the mid 1960s a very beautiful theory was in place, showing that one could recognize positivity geometrically, cohomologically, or numerically. During the same years, Zariski and others began to investigate the more complicated be- havior of linear series defined by line bundles that may not be ample. -
Resonance Varieties Over Fields of Positive Characteristic
Resonance varieties over fields of positive characteristic Michael J. Falk July 11, 2018 Abstract Let A be a hyperplane arrangement, and k a field of arbitrary char- acteristic. We show that the projective degree-one resonance variety R1(A, k) of A over k is ruled by lines, and identify the underlying algebraic line complex L(A, k) in the Grassmannian G(2, kn), n = |A|. L(A, k) is a union of linear line complexes corresponding to the neighborly partitions of subarrangements of A. Each linear line complex is the intersection of a family of special Schubert varieties corresponding to a subspace arrange- ment determined by the partition. In case k has characteristic zero, the resulting ruled varieties are lin- ear and pairwise disjoint, by results of A. Libgober and S. Yuzvinsky. We give examples to show that each of these properties fails in positive char- acteristic. The (4,3)-net structure on the Hessian arrangement gives rise 1 4 to a nonlinear component in R (A, Z3), a cubic hypersurface in P with interesting line structure. This provides a negative answer to a question of A. Suciu. The deleted B3 arrangement has linear resonance components over Z2 that intersect nontrivially. 1 Resonance and characteristic varieties Arising out of the study of local system cohomology and fundamental groups, arXiv:math/0611008v1 [math.CO] 1 Nov 2006 characteristic and resonance varieties of complex hyperplane arrangements have become the object of much of the current research in the field. The study of resonance varieties in particular has led to surprising connections with other areas of mathematics: generalized Cartan matrices [23], Latin squares and loops, nets [40], special pencils of plane curves [23, 13], graded resolutions and the BGG correspondence [10, 33, 34], and the Bethe Ansatz [7].