Algebraic Surfaces in Positive Characteristic
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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. -
1 Real-Time Algebraic Surface Visualization
1 Real-Time Algebraic Surface Visualization Johan Simon Seland1 and Tor Dokken2 1 Center of Mathematics for Applications, University of Oslo [email protected] 2 SINTEF ICT [email protected] Summary. We demonstrate a ray tracing type technique for rendering algebraic surfaces us- ing programmable graphics hardware (GPUs). Our approach allows for real-time exploration and manipulation of arbitrary real algebraic surfaces, with no pre-processing step, except that of a possible change of polynomial basis. The algorithm is based on the blossoming principle of trivariate Bernstein-Bezier´ func- tions over a tetrahedron. By computing the blossom of the function describing the surface with respect to each ray, we obtain the coefficients of a univariate Bernstein polynomial, de- scribing the surface’s value along each ray. We then use Bezier´ subdivision to find the first root of the curve along each ray to display the surface. These computations are performed in parallel for all rays and executed on a GPU. Key words: GPU, algebraic surface, ray tracing, root finding, blossoming 1.1 Introduction Visualization of 3D shapes is a computationally intensive task and modern GPUs have been designed with lots of computational horsepower to improve performance and quality of 3D shape visualization. However, GPUs were designed to only pro- cess shapes represented as collections of discrete polygons. Consequently all other representations have to be converted to polygons, a process known as tessellation, in order to be processed by GPUs. Conversion of shapes represented in formats other than polygons will often give satisfactory visualization quality. However, the tessel- lation process can easily miss details and consequently provide false information. -
On the Euler Number of an Orbifold 257 Where X ~ Is Embedded in .W(X, G) As the Set of Constant Paths
Math. Ann. 286, 255-260 (1990) Imam Springer-Verlag 1990 On the Euler number of an orbifoid Friedrich Hirzebruch and Thomas HSfer Max-Planck-Institut ffir Mathematik, Gottfried-Claren-Strasse 26, D-5300 Bonn 3, Federal Republic of Germany Dedicated to Hans Grauert on his sixtieth birthday This short note illustrates connections between Lothar G6ttsche's results from the preceding paper and invariants for finite group actions on manifolds that have been introduced in string theory. A lecture on this was given at the MPI workshop on "Links between Geometry and Physics" at Schlol3 Ringberg, April 1989. Invariants of quotient spaces. Let G be a finite group acting on a compact differentiable manifold X. Topological invariants like Betti numbers of the quotient space X/G are well-known: i a 1 b,~X/G) = dlmH (X, R) = ~ g~)~ tr(g* I H'(X, R)). The topological Euler characteristic is determined by the Euler characteristic of the fixed point sets Xa: 1 Physicists" formula. Viewed as an orbifold, X/G still carries someinformation on the group action. In I-DHVW1, 2; V] one finds the following string-theoretic definition of the "orbifold Euler characteristic": e X, 1 e(X<~.h>) Here summation runs over all pairs of commuting elements in G x G, and X <g'h> denotes the common fixed point set ofg and h. The physicists are mainly interested in the case where X is a complex threefold with trivial canonical bundle and G is a finite subgroup of $U(3). They point out that in some situations where X/G has a resolution of singularities X-~Z,X/G with trivial canonical bundle e(X, G) is just the Euler characteristic of this resolution [DHVW2; St-W]. -
Algebraic Curves and Surfaces
Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces . -
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. -
Automorphisms of Minimal Entropy on Supersingular K3 Surfaces
AUTOMORPHISMS OF MINIMAL ENTROPY ON SUPERSINGULAR K3 SURFACES SIMON BRANDHORST, VÍCTOR GONZÁLEZ-ALONSO Abstract. In this article we give a strategy to decide whether the logarithm of a given Salem number is realized as entropy of an automorphism of a su- persingular K3 surface in positive characteristic. As test case it is proved that log λd, where λd is the minimal Salem number of degree d, is realized in char- acteristic 5 if and only if d ≤ 22 is even and d 6= 18. In the complex projective setting we settle the case of entropy log λ12, left open by McMullen, by giv- ing the construction. A necessary and sufficient test is developed to decide whether a given isometry of a hyperbolic lattice, with spectral radius bigger than one, is positive, i.e. preserves a chamber of the positive cone. Contents 1. Introduction 1 Notation 4 Acknowledgements 4 2. Lattices 4 3. Torelli theorems for supersingular K3 surfaces 5 4. Positivity 8 5. Lattices in number fields 11 6. Gluing lattices and isometries 12 7. Realizability Conditions 14 8. Realized Salem numbers 18 Appendix A. Realizing λ12 over C 25 References 25 arXiv:1609.02716v2 [math.AG] 20 Jul 2017 1. Introduction A Salem number is a real algebraic integer λ > 1 which is conjugate to 1/λ and whose other conjugates lie on the unit circle. In each even degree d there is a unique smallest Salem number λd. Conjecturally the smallest Salem number is λ10, found by Lehmer in 1933 [17]. Date: September, 2016. 2010 Mathematics Subject Classification. -
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. -
Mathematisches Forschungsinstitut Oberwolfach Moduli Spaces And
Mathematisches Forschungsinstitut Oberwolfach Report No. 23/2016 DOI: 10.4171/OWR/2016/23 Moduli spaces and Modular forms Organised by Jan Hendrik Bruinier, Darmstadt Gerard van der Geer, Amsterdam Valery Gritsenko, Villeneuve d’Ascq 24 April – 30 April 2016 Abstract. The roots of both moduli spaces and modular forms go back to the theory of elliptic curves in the 19th century. Both topics have seen an enormous growth in the second half of the 20th century, but the interaction between the two remained limited. Recently there have been new develop- ments that led to new points of contact between the two topics. One is the theory of K3 surfaces that is rapidly gaining a lot of new interest. Here the link with modular forms on orthogonal groups has led to progress on the Kodaira dimension of the moduli spaces of K3 surfaces. Another new devel- opment has been the use of moduli spaces of curves to gather new information about Siegel modular forms. The workshop intended to bring representatives from both the theory of moduli and the theory of modular forms together to further the interaction between the two topics as the time seemed ripe to do this. Mathematics Subject Classification (2010): 11xx, 14xx. Introduction by the Organisers The workshop Moduli Spaces and Modular Forms, organized by Jan Bruinier (Darmstadt), Gerard van der Geer (Amsterdam) and Val´ery Gritsenko (Lille) was held 25-29 April, 2016 and was attended by 52 participants from all over the world. The attendance ranged from senior leaders in the field to young postdocs and advanced Ph.D. -
Supersingular K3 Crystals Astérisque, Tome 64 (1979), P
Astérisque ARTHUR OGUS Supersingular K3 crystals Astérisque, tome 64 (1979), p. 3-86 <http://www.numdam.org/item?id=AST_1979__64__3_0> © Société mathématique de France, 1979, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ SUPERSINGULAR K3 CRYSTALS by Arthur OGUS (Berkeley) §0. INTRODUCTION. This paper is intented as propaganda for the machinery of crystalline cohomolo- gy, and in particular for the philosophy that F-crystals are a partial analogue, in characteristic p, to Hodge structures in characteristic zero. An extremely rudimen tary start along this road, for "abstract" F-crystals and Hodge structures, was made in [15] ; here we turn to crystals arising geometrically, especially from supersingu- lar abelian varieties and K3 surfaces. As we shall see, it is reasonable to hope that the moduli of such varieties are given by the moduli of their F-crystals, which in fact form explicit "period-spaces". Here is a plan of the paper : The first section contains some refinements of gene rally known facts concerning crystalline Chern classes, e.g. an integral version of Bloch's theorem relating flat and crystalline cohomology (1.7), Conditions guaran- teeing that c : Pic <8> 2£/p2£ H,™ is injective (1.4), and a formula for certain second order obstructions to extending invertible sheaves in a family (l.