Introduction to Algebraic Surfaces Lecture Notes for the Course at the University of Mainz
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Arithmetic of the Moduli of Fibered Algebraic Surfaces with Heuristics for Counting Curves Over Global Fields
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Minnesota Digital Conservancy Arithmetic of the moduli of fibered algebraic surfaces with heuristics for counting curves over global fields A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Jun Yong Park IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy Craig Westerland May, 2018 © Jun Yong Park 2018 ALL RIGHTS RESERVED Acknowledgements First and foremost, I would like to express sincere gratitude to my doctoral advisor Craig Westerland for his kind guidance throughout the theme of studying arithmetic of moduli spaces under the interplay of geometry & topology with number theory. His insights and suggestions have helped me grow as a researcher and as a mathematician. I would also like to thank Denis Auroux, Jordan Ellenberg, Benson Farb, Thomas Espitau, Robert Gulliver, Joe Harris, Changho Han, Leo Herr, Minhyong Kim, Anatole Katok, Maxim Kazarian, Honglei Lang, Frank Morgan, Steven Sperber, Dennis Stan- ton, Vladimir Sverak, András Stipsicz, Sergei Tabachnikov, Yu-jong Tzeng, Alexander Voronov, Chuen-Ming Michael Wong, Jonathan Wise and Jesse Wolfson for helpful conversations, numerous suggestions and inspirations. A majority this thesis was written during my stay at the Brown University, Depart- ment of Mathematics in January-July 2018. I would like to thank Department Chair Dan Abramovich and Graduate Advisor Thomas Goodwillie as well as Dori Bejleri, Ali- cia Harper, Giovanni Inchiostro and many other wonderful graduate students at Brown for their kind hospitality. Also a part of this thesis work was done during my stay at IBS-CGP in July-August 2017, I would like to thank Yong-Geun Oh, Jae-Suk Park, and Jihun Park for their kind hospitality. -
Algebraic Surfaces with Minimal Betti Numbers
P. I. C. M. – 2018 Rio de Janeiro, Vol. 2 (717–736) ALGEBRAIC SURFACES WITH MINIMAL BETTI NUMBERS JH K (금종해) Abstract These are algebraic surfaces with the Betti numbers of the complex projective plane, and are called Q-homology projective planes. Fake projective planes and the complex projective plane are smooth examples. We describe recent progress in the study of such surfaces, singular ones and fake projective planes. We also discuss open questions. 1 Q-homology Projective Planes and Montgomery-Yang problem A normal projective surface with the Betti numbers of the complex projective plane CP 2 is called a rational homology projective plane or a Q-homology CP 2. When a normal projective surface S has only rational singularities, S is a Q-homology CP 2 if its second Betti number b2(S) = 1. This can be seen easily by considering the Albanese fibration on a resolution of S. It is known that a Q-homology CP 2 with quotient singularities (and no worse singu- larities) has at most 5 singular points (cf. Hwang and Keum [2011b, Corollary 3.4]). The Q-homology projective planes with 5 quotient singularities were classified in Hwang and Keum [ibid.]. In this section we summarize progress on the Algebraic Montgomery-Yang problem, which was formulated by J. Kollár. Conjecture 1.1 (Algebraic Montgomery–Yang Problem Kollár [2008]). Let S be a Q- homology projective plane with quotient singularities. Assume that S 0 := S Sing(S) is n simply connected. Then S has at most 3 singular points. This is the algebraic version of Montgomery–Yang Problem Fintushel and Stern [1987], which was originated from pseudofree circle group actions on higher dimensional sphere. -
Hodge Theory and Moduli
Contemporary Mathematics Volume 766, 2021 https://doi.org/10.1090/conm/766/15381 Hodge theory and moduli Phillip Griffiths Outline I.A Introductory comments I.B Classification problem; first case of relation between moduli and Hodge theory II.A Moduli II.B Hodge theory III.A Generalities on Hodge theory and moduli III.B I-surfaces and their period mappings Appendix; Application of Hodge theory to I-surfaces with a degree 2 el- liptic singularity I.A. Introductory comments. Algebraic geometry is frequently seen as a very interesting and beautiful subject, but one that is also very difficult to get into. This is partly due to its breadth, as traditionally algebra (commutative and homological), topology, analysis, differential geometry, Lie theory — and more recently combinatorics, logic, categorical and derived algebraic techniques, . — are used to study it. I have tried to make these notes accessible to a general audience by reviewing some of the most basic concepts, illustrating the material with elementary examples and informal geometric and heuristic arguments, and with occasional side comments for experts in the subject. Although algebra, both commutative and homological, are the central tools in algebraic geometry, their use in many current areas of research (birational geometry and the minimal model program) is frequently quite technical and will not be extensively discussed in these notes. On the other hand, partly because Hodge theory involves analysis, Lie theory and differential geometry as well as a wide variety of homological methods, it is perhaps less prevalent in the more algebraic works in the field. One objective of these talks is to illustrate how, in partnership with the more algebraic and homological methods, Hodge theory may be used to study interesting and important geometric questions. -
Workshop on Syzygies in Algebraic Geometry, with an Exploration of a Connection with String Theory
Workshop on Syzygies in Algebraic Geometry, with an exploration of a connection with String Theory Lawrence Ein (University of Illinois at Chigago), David Eisenbud (University of California, Berkeley), Gavril Farkas (Humboldt Universitat¨ zu Berlin), Irena Peeva (Cornell University) August 12–17, 2012 1 Overview of the Field Free resolutions are often naturally attached to geometric objects. A question of prime interest has been to understand what constraints the geometry of a variety imposes on the corresponding Betti numbers and structure of the resolution. In recent years, free resolutions have been also impressively used to study the birational geometry of moduli spaces of curves. One of the most exciting and challenging long-standing open conjectures on free resolutions is the Regu- larity Eisenbud-Goto Conjecture that the regularity of a prime ideal is bounded above by its multiplicity. The conjecture has roots in Castelnuovo’s work, and is known to hold in only a few cases: the Cohen-Macaulay case, for curves, and for smooth surfaces. In the 80’s, Mark Green [11], [12] conjectured that the the minimal resolution of the canonical ring a non-hyperelliptic curve X of genus g satisfies the Np property if and only if the Clifford index of X is g−1 greater than P . Recall that N1 says that the homogenous ideal of X in P is generated by quadrics. For p ≥ 2, Np means that the first p − 1 steps of the minimal resolution of the homogenous ideal of X are given by matrices of linear forms. Using geometry of the Hilbert schemes of K3 surfaces, Voisin [20], [21] showed that the Green conjecture holds for generic curves. -
Report No. 26/2008
Mathematisches Forschungsinstitut Oberwolfach Report No. 26/2008 Classical Algebraic Geometry Organised by David Eisenbud, Berkeley Joe Harris, Harvard Frank-Olaf Schreyer, Saarbr¨ucken Ravi Vakil, Stanford June 8th – June 14th, 2008 Abstract. Algebraic geometry studies properties of specific algebraic vari- eties, on the one hand, and moduli spaces of all varieties of fixed topological type on the other hand. Of special importance is the moduli space of curves, whose properties are subject of ongoing research. The rationality versus general type question of these spaces is of classical and also very modern interest with recent progress presented in the conference. Certain different birational models of the moduli space of curves have an interpretation as moduli spaces of singular curves. The moduli spaces in a more general set- ting are algebraic stacks. In the conference we learned about a surprisingly simple characterization under which circumstances a stack can be regarded as a scheme. For specific varieties a wide range of questions was addressed, such as normal generation and regularity of ideal sheaves, generalized inequalities of Castelnuovo-de Franchis type, tropical mirror symmetry constructions for Calabi-Yau manifolds, Riemann-Roch theorems for Gromov-Witten theory in the virtual setting, cone of effective cycles and the Hodge conjecture, Frobe- nius splitting, ampleness criteria on holomorphic symplectic manifolds, and more. Mathematics Subject Classification (2000): 14xx. Introduction by the Organisers The Workshop on Classical Algebraic Geometry, organized by David Eisenbud (Berkeley), Joe Harris (Harvard), Frank-Olaf Schreyer (Saarbr¨ucken) and Ravi Vakil (Stanford), was held June 8th to June 14th. It was attended by about 45 participants from USA, Canada, Japan, Norway, Sweden, UK, Italy, France and Germany, among of them a large number of strong young mathematicians. -
SINTESI DELLA TESI Enriques-Kodaira Classification Of
Universit`adegli Studi Roma Tre Facolta` di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Matematica SINTESI DELLA TESI Enriques-Kodaira classification of Complex Algebraic Surfaces Candidato Relatore Cristian Minoccheri Prof.ssa Lucia Caporaso Anno Accademico 2011-12 Sessione di Luglio 2012 Our principal aim is to describe and prove the classification of complex algebraic surfaces, a result due to the Italian school of Algebraic Geometry at the beginning of the XX century, and in particular to Enriques. This classification is done up to birational maps, but it is different from the classi- fication of curves. In fact, in the case of curves there is a unique nonsingular projective model for each equivalence class, whereas in the case of surfaces this model is not unique. Therefore we will need to deal with something dif- ferent: the minimal models. In fact, apart from the case of ruled surfaces (i.e. those birational to the product of a curve and the projective line), these are unique and allow us to obtain birationally equivalent surfaces by successive blow-ups. Thus the classification problem will split in two parts: ruled surfaces, which require special considerations, and non-ruled ones, for which it will suffice to classify minimal models. For this, we will need some birational invariants, which capture the geometric peculiarities of each class, as we will see. A first rough classification is achieved by means of Kodaira dimension, κ, which is defined as the largest dimension of the image of the surface in a projective space by the rational map determined by the linear system jnKj, or as −1 if jnKj = ? for every n. -
[Math.AG] 1 Oct 2002 Nt,O Neaecover Etale an Or finite, Surface Iia N O Emnlsurface Terminal Log and Minimal Rewhen True Exist
ALGEBRAIC SURFACES WITH QUOTIENT SINGULARITIES - INCLUDING SOME DISCUSSION ON AUTOMORPHISMS AND FUNDAMENTAL GROUPS J. Keum and D. -Q. Zhang Introduction We work over the complex numbers field C. In the present survey, we report some recent progress on the study of varieties with mild singularities like log terminal singularities (which are just quotient singularities in the case of dimension 2; see [KMM]). Singularities appear naturally in many ways. The minimal model program developed by Mori et al shows that a minimal model will inevitably have some terminal singularities [KMo]. Also the degenerate fibres of a family of varieties will have some singularities. We first follow Iitaka’s strategy to divide (singular) varieties Y according to the loga- rithmic Kodaira dimension κ(Y 0) of the smooth locus Y 0 of Y . One key result (2.3) says that for a relatively minimal log terminal surface Y we have either nef KY or dominance of Y 0 by an affine-ruled surface. It is conjectured to be true for any dimension [KMc]. In smooth projective surfaces of general type case, we have Miyaoka-Yau inequality 2 2 2 c1 3c2 and Noether inequalities: pg (1/2)c1 +2, c1 (1/5)c2 (36/5). Similar inequalities≤ are given for Y 0 in Section 4;≤ these will give effective≥ restriction− on the region where non-complete algebraic surfaces of general type exist. In Kodaira dimension zero case, an interesting conjecture (3.12) (which is certainly true when Y is smooth projective by the classification theory) claims that for a relatively 0 0 minimal and log terminal surface Y of Kodaira dimension κ(Y ) = 0, one has either π1(Y ) finite, or an etale cover Z0 Y 0 where Z0 is the complement of a finite set in an abelian surface Z. -
Six Lectures on Compact Complex Surfaces
SIX LECTURES ON COMPACT COMPLEX SURFACES ALESSANDRA SARTI Abstract. This is a formulation of my six lectures (about 12h) on Superficies Algebraicas Complejas during my stay at the UBA, the university of Buenos Aires in june/july 2010. I would like to thank all the students and collegues attending my lectures. Particular thanks go to my hosts Fernando Cukiermann and Alicia Dickenstein, to Armando Treibich and the PREMER programm making my stay at UBA possible. 1 Contents 1. Introduction 2 1.1. Classify 2 1.2. Curves 3 1.3. Surfaces 4 2. ThePicardGroupandtheRiemann–Rochtheorem 5 2.1. The Picard group 5 2.2. Intersection theory on surfaces 6 2.3. TheRiemann-Rochtheoremforsurfaces 9 3. Birational Geometry 12 3.1. Rational maps and linear systems 12 3.2. Blow-up of a point 14 3.3. Indeterminacies 16 3.4. Castelnuovo’s contraction theorem 18 3.5. Kodaira dimension 19 Date: January 4, 2015. 1 A5 × A5-symmetric surface of degree 12 with 600 nodes 1 2 Alessandra Sarti 4. Kodaira dimension −∞ 21 4.1. Ruled and rational surfaces 21 4.2. Invariantsof(geometrically)ruledsurfaces 22 4.3. Rational surfaces 23 4.4. Minimal ruled surfaces 23 4.5. Some example of rational surfaces 26 4.6. Linear system of conics 26 4.7. Linear system of cubics 26 5. Kodaira dimension 0 27 5.1. Unicity of the minimal model 27 5.2. K3 surfaces, Enriques surfaces 28 References 29 1. Introduction 1.1. Classify. When studying a geometric problem, one has: • to define the family of objects one wants to investigate, • to define the notion of equivalence between the objects, • to introduce invariants to understand if the objects are equivalent. -
The Moduli of Curves of Genus Six and K3 Surfaces
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 3, March 2011, Pages 1445–1462 S 0002-9947(2010)05126-8 Article electronically published on October 25, 2010 THE MODULI OF CURVES OF GENUS SIX AND K3 SURFACES MICHELA ARTEBANI AND SHIGEYUKI KONDO¯ Abstract. We prove that the coarse moduli space of curves of genus six is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized K3 surfaces. Introduction This paper gives a birational period map between the coarse moduli space M6 of curves of genus six and a bounded symmetric domain of type IV, parametrizing some lattice-polarized K3 surfaces. A similar correspondence, between a moduli space of curves and a period space of K3 surfaces, was given by the second author for curves of genus three and genus four in [Ko2] and [Ko3]. Some of the results in this paper were announced in [Ko3]. Realizing a moduli space as an arithmetic quotient of a bounded symmetric domain is interesting for several reasons. The first reason is that this gives a connection with the theory of automorphic forms. Recently Borcherds [Bor] gave a systematic method to construct automor- phic forms on a bounded symmetric domain of type IV. Allcock and Freitag [AF] and the second author [Ko4], [Ko5] used his theory to obtain explicit embeddings in a projective space for the moduli spaces of cubic surfaces, Enriques surfaces and curves of genus three, respectively. Secondly, an arithmetic quotient of a bounded symmetric domain has several compactifications, that is, Satake-Baily-Borel’s, Mumford’s toroidal and Looijenga’s compactifications [Lo1], [Lo2]. -
Stable Rationality of Quadric and Cubic Surface Bundle Fourfolds
STABLE RATIONALITY OF QUADRIC AND CUBIC SURFACE BUNDLE FOURFOLDS ASHER AUEL, CHRISTIAN BOHNING,¨ AND ALENA PIRUTKA Abstract. We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the degeneration method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X 2 3 of bidegree (2; 3) in P × P is not stably rational. Via projections onto the two 2 3 factors, X ! P is a cubic surface bundle and X ! P is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, 2 we introduce, for any n ≥ 4, new quadric surface bundle fourfolds Xn ! P with 2 discriminant curve Dn ⊂ P of degree 2n, such that Xn has nontrivial unramified Brauer group and admits a universally CH0-trivial resolution. 1. Introduction m An integral variety X over a field k is stably rational if X × P is rational, for some m. In recent years, failure of stable rationality has been established for many classes of smooth rationally connected projective complex varieties, see, for instance [1,3,4,7, 15, 16, 22, 23, 24, 25, 26, 27, 28, 29, 30, 33, 34, 36, 37]. These results were obtained by the specialization method, introduced by C. Voisin [37] and developed in [16]. In many applications, one uses this method in the following form. For simplicity, assume that k is an uncountable algebraically closed field and consider a quasi-projective integral scheme B over k and a generically smooth projective morphism X! B with positive dimensional fibers. -
The Cohomology of Coherent Sheaves
CHAPTER VII The cohomology of coherent sheaves 1. Basic Cechˇ cohomology We begin with the general set-up. (i) X any topological space = U an open covering of X U { α}α∈S a presheaf of abelian groups on X. F Define: (ii) Ci( , ) = group of i-cochains with values in U F F = (U U ). F α0 ∩···∩ αi α0,...,αYi∈S We will write an i-cochain s = s(α0,...,αi), i.e., s(α ,...,α ) = the component of s in (U U ). 0 i F α0 ∩··· αi (iii) δ : Ci( , ) Ci+1( , ) by U F → U F i+1 δs(α ,...,α )= ( 1)j res s(α ,..., α ,...,α ), 0 i+1 − 0 j i+1 Xj=0 b where res is the restriction map (U U ) (U U ) F α ∩···∩ Uαj ∩···∩ αi+1 −→ F α0 ∩··· αi+1 and means “omit”. Forb i = 0, 1, 2, this comes out as δs(cα , α )= s(α ) s(α ) if s C0 0 1 1 − 0 ∈ δs(α , α , α )= s(α , α ) s(α , α )+ s(α , α ) if s C1 0 1 2 1 2 − 0 2 0 1 ∈ δs(α , α , α , α )= s(α , α , α ) s(α , α , α )+ s(α , α , α ) s(α , α , α ) if s C2. 0 1 2 3 1 2 3 − 0 2 3 0 1 3 − 0 1 2 ∈ One checks very easily that the composition δ2: Ci( , ) δ Ci+1( , ) δ Ci+2( , ) U F −→ U F −→ U F is 0. Hence we define: 211 212 VII.THECOHOMOLOGYOFCOHERENTSHEAVES s(σβ0, σβ1) defined here U σβ0 Uσβ1 Vβ1 Vβ0 ref s(β0, β1) defined here Figure VII.1 (iv) Zi( , ) = Ker δ : Ci( , ) Ci+1( , ) U F U F −→ U F = group of i-cocycles, Bi( , ) = Image δ : Ci−1( , ) Ci( , ) U F U F −→ U F = group of i-coboundaries Hi( , )= Zi( , )/Bi( , ) U F U F U F = i-th Cech-cohomologyˇ group with respect to . -
Finite Generation of a Canonical Ring
Finite generation of a canonical ring Yujiro Kawamata February 21, 2013 Abstract The purpose of this note is to review an algebraic proof of the finite generation theorem due to Birkar-Cascini-Hacon-McKernan [5] whose method is based on the Minimal Model Program (MMP). An analytic proof by Siu [57] will be reviewed by Mihai Paun. 1 Introduction The finite generation of canonical rings was a problem considered by Zariski [64], and the proof in the case of dimension 2 due to Mumford [41] in the appendix of [64] is one of the motivations towards the minimal model theory of higher dimensional algebraic varieties. Let X be a smooth projective variety defined over a field k, D a divisor on X, and OX (D) the associated invertible sheaf. Many problems in algebraic geometry are translated into questions on the vector space of holomorphic 0 0 sections H (X, D) = H (X, OX (D)). The Riemann-Roch problem is to de- termine this vector space. For example, the Riemann-Roch theorem tells arXiv:0804.3151v1 [math.AG] 19 Apr 2008 n p p us that the alternating sum p=0(−1) dim H (X, OX (D)) is expressed in terms of topological invariants.P Instead of considering a single vector space, we look at the graded ring ∞ R(X, D)= H0(X, mD) mM=0 called the section ring for the pair (X, D), where we use the additive notation for divisors instead of multiplicative one for sheaves; we have OX (mD) = 1 ⊗m OX (D) . There are obvious multiplication homomorphisms 0 0 0 H (X, m1D) ⊗ H (X, m2D) → H (X, (m1 + m2)D) and the section ring becomes a graded algebra over the base field k = 0 H (X, OX ).