Enumerative Geometry Steven Finch. February 24, 2014
|
|
- Dortha Sullivan
- 5 years ago
- Views:
Transcription
1 Enumerative Geometry Steven Finch February 24, 204 Given a complex projective variety V (as defined in []), we wish to count the curves in V that satisfy certain prescribed conditions. Let C f denote complex projective -dimensional space. In our first example, V = C f2, the complex projective plane; in the second and third, V is a general hypersurface in C f of degree 2 3. Call such V a cubic twofold when =3and a quintic threefold when =4. Our interest is in rational curves, which include all lines (degree ), conics (degree 2) and singular cubics (degree 3). No elliptic curves are rational. The word rational here refers to the affine parametrization of the curve a ratio of polynomials andthecurveisofdegree if the polynomials are of degree at most. For instance, the circle =is represented as = 2 + = The lemniscate of Bernoulli has degree 4 andisrepresentedas 4 = = 2 ( 2 ) It is also defined implicitly: = 2 2 The semi- and clearly possesses a singularity (vanishing gradient) at the origin. cubical parabola 2 = 3 and four-petal rose =4 2 2 possess likewise. All rational curves, smooth or not, have genus 0. 0 Copyright c 204 by Steven R. Finch. All rights reserved.
2 Enumerative Geometry Rational Plane Curves Passing Through Points. In the following, we use homogeneous coordinates. Given two distinct points ( ), ( ) in fc 2, there is exactly one line passing through bothbecausethesimultaneoussystem of equations + + =0 { 2} has a unique solution ( ) in C f2 (up to a common scalar). It is a little harder to prove the corresponding result for conics. Given five points ( ) in general position, there is exactly one conic passing through all five via study of =0 { } in C f5. Hence we have = 2 =,where is defined as the number of rational curves in C f2 of degree passing through 3 general points. The quantity 3 turns out to be the critical threshold for our question: less would give an answer of infinity, more would give an answer of zero [2]. Proving that 3 =2involves a heavy dose of algebraic geometry [3, 4]. Credit for this accomplishment (in the mid-800s) is assigned variously to Chasles [5] and Steiner [6]. Kontsevich s famous recursion [7, 8, 9]: = X µ µ = 2 was not found until recently (in 994). Its astonishing proof drew upon ideas not from geometry but from mathematical physics, specifically, quantum field theory and string theory. Other relevant recursions for curve counting are known [7, 0,, 2] but these are too complicated for us to discuss here. The asymptotics for are [, 3] µ (3 )! ( ) as, obtained using a device due to Zagier called the asymp trick. closed-form expression for these constants is known. No 0.2. Lines On a Hypersurface. The fact that exactly 27 lines lie on a cubic twofold in f C 3 is a well-known theorem [4, 5] due to Cayley & Salmon (in 849). Somewhat later, Schubert proved (in 886) that exactly 2875 lines lie on a quintic threefold in f C 4.Thuswehave 3 =27and 4 =2875,where is defined as the
3 Enumerative Geometry 3 number of lines on a general hypersurface in C f of degree 2 3. Expanding on these results, van der Waerden proved (in 933) that à 2 3 Y = ( ) (2 3 + )! ( )! and Zagier [9, 3] obtained asymptotics r µ 27 (2 3) as. In this case, closed-form expressions are available Rational Curves On a Quintic Threefold. The number of conics on a cubic twofold is infinity. Incontrast,thenumberofconicsonaquinticthreefold is Our discussion at this point becomes highly speculative it is merely conjectured (by Clemens [8]) that the number of degree rational curves on a quintic threefold is finite but the following calculations are known to be valid at least for 9. Define 0 (), (), 2 () via power series expansion of a certain hypergeometric function [4]: Q 5 = (5 + ) Q = (5 + = 0()+ () + 2 () 2 + )5 It follows that 0 () = (5)! (!) 5 () = à (5)! (!) 5 5X =+! (a similar expression for 2 () would be good to see). We then define rational numbers recursively from yielding 2 () = () () + 5 ½ { } = = µ 0 ()exp () 0 () ¾
4 Enumerative Geometry 4 Such numbers are examples of Gromov-Witten invariants, which count not only the rational curves we desire, but also capture (unwanted) additional structure [8]. The final step is another recursion [4, 6]: = X 3 yielding { } = = { } It is, again, merely conjectured (by Gopakumar & Vafa [8, 9]) that all numbers obtained in this manner are indeed integers. Much work lies ahead to rigorously confirm everything written here. The asymptotics for remain open Addendum. Let be a cubic twofold and let be the number of rational curves on of degree passing through general points on. Traves [9, 7] gave the values { } = = { } and conjectured that is always finite. A recursive formula for (à la Kontsevich for?) remains open. References [] S. R. Finch, Elliptic curves over Q, unpublished note (2005). [2] I. Strachan, How to count curves: from nineteenth-century problems to twentyfirst-century solutions, RoyalSoc.Lond.Philos.Trans.Ser.A36 (2003) ; MR (2005c:53). [3] L. Caporaso, Counting curves on surfaces: a guide to new techniques and results, European Congress of Mathematics, v.i,proc.996budapestconf.,ed.a.balog, G. O. H. Katona, A. Recski and D. Szász, Birkhäuser, 998, pp ; arxiv:alg-geom/96029; MR (99j:4050). [4] A. Gathmann, Enumerative geometry, [5] E. Arbarello, M. Cornalba and P. A. Griffiths, Geometry of Algebraic Curves, v. II, Springer-Verlag, 20, pp ; MR (202e:4059). [6] S. Lanzat and M. Polyak, Counting real curves with passage/tangency conditions, J. London Math. Soc. 85 (202) ; arxiv:0.686; MR
5 Enumerative Geometry 5 [7] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys. 64 (994) ; arxiv:hepth/940247; MR29244 (95i:4049). [8] S. Katz, Enumerative Geometry and String Theory, Amer. Math. Soc., 2006, pp. 0, 27 3, 25 43, 92 96; MR (2007i:4054). [9] N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences, A03587, A027363, A06004, A07692, and A [0] C. Itzykson, Counting rational curves on rational surfaces, Internat.J.Modern Phys. B 8 (994) ; MR30568 (96e:406). [] P. Di Francesco and C. Itzykson, Quantum intersection rings, The Moduli Space of Curves, Proc. 994 Texel Island conf., ed. R. Dijkgraaf, C. Faber and G. van der Geer, Birkhäuser Boston, 995, pp. 8 48; arxiv:hep-th/94275v; MR (96k:404a). [2] P. Di Francesco, Enumerative geometry from string theory, Math. Comput. Modelling 26 (997) 75 96; MR (98j:4072). [3] D. B. Grünberg and P. Moree, Sequences of enumerative geometry: congruences and asymptotics, Experim. Math. 7 (2008) ; arxiv:math/060286; MR (200k:49). [4] J. J. O Connor and E. F. Robertson, Cubic surfaces, [5] B. Hunt, TheGeometryofSomeSpecialArithmeticQuotients, Springer-Verlag, 996, pp , ; MR (98c:4033). [6] M. Kontsevich, Enumeration of rational curves via torus actions. The Moduli Space of Curves, Proc. 994 Texel Island conf., ed. R. Dijkgraaf, C. Faber and G. van der Geer, Birkhäuser Boston, 995, pp ; arxiv:hep-th/ ; MR (97d:4077). [7] W. Traves, Conics on the cubic surface,
Introduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway
Curves and surfaces Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway What is algebraic geometry? IMA, April 13, 2007 Outline Introduction Curves Surfaces Curves on surfaces
More informationCounting curves on a surface
Counting curves on a surface Ragni Piene Centre of Mathematics for Applications and Department of Mathematics, University of Oslo University of Pennsylvania, May 6, 2005 Enumerative geometry Specialization
More informationON THE USE OF PARAMETER AND MODULI SPACES IN CURVE COUNTING
ON THE USE OF PARAMETER AND MODULI SPACES IN CURVE COUNTING RAGNI PIENE In order to solve problems in enumerative algebraic geometry, one works with various kinds of parameter or moduli spaces:chow varieties,
More informationCOUNTING ELLIPTIC PLANE CURVES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 12, December 1997, Pages 3471 3479 S 0002-9939(97)04136-1 COUNTING ELLIPTIC PLANE CURVES WITH FIXED j-invariant RAHUL PANDHARIPANDE (Communicated
More informationFrom de Jonquières Counts to Cohomological Field Theories
From de Jonquières Counts to Cohomological Field Theories Mara Ungureanu Women at the Intersection of Mathematics and High Energy Physics 9 March 2017 What is Enumerative Geometry? How many geometric structures
More informationEnumerative Geometry: from Classical to Modern
: from Classical to Modern February 28, 2008 Summary Classical enumerative geometry: examples Modern tools: Gromov-Witten invariants counts of holomorphic maps Insights from string theory: quantum cohomology:
More informationCurve counting and generating functions
Curve counting and generating functions Ragni Piene Università di Roma Tor Vergata February 26, 2010 Count partitions Let n be an integer. How many ways can you write n as a sum of two positive integers?
More informationarxiv: v1 [math.ag] 15 Dec 2013
INTEGRALITY OF RELATIVE BPS STATE COUNTS OF TORIC DEL PEZZO SURFACES arxiv:1312.4112v1 [math.ag] 15 Dec 2013 MICHEL VAN GARREL, TONY W. H. WONG, AND GJERGJI ZAIMI Abstract. Relative BPS state counts for
More informationPROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013
PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013 1. Problems on moduli spaces The main text for this material is Harris & Morrison Moduli of curves. (There are djvu files
More informationOn the intersection theory of the moduli space of rank two bundles
On the intersection theory of the moduli space of rank two bundles Alina Marian a, Dragos Oprea b,1 a Yale University, P.O. Box 208283, New Haven, CT, 06520-8283, USA b M.I.T, 77 Massachusetts Avenue,
More informationBjorn Poonen. MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional Varieties. January 17, 2006
University of California at Berkeley MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional (organized by Jean-Louis Colliot-Thélène, Roger Heath-Brown, János Kollár,, Alice Silverberg,
More informationGeometric Chevalley-Warning conjecture
Geometric Chevalley-Warning conjecture June Huh University of Michigan at Ann Arbor June 23, 2013 June Huh Geometric Chevalley-Warning conjecture 1 / 54 1. Chevalley-Warning type theorems June Huh Geometric
More informationPeriods and generating functions in Algebraic Geometry
Periods and generating functions in Algebraic Geometry Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil www.impa.br/ hossein/ Abstract In 1991 Candelas-de la Ossa-Green-Parkes predicted
More informationOverview of classical mirror symmetry
Overview of classical mirror symmetry David Cox (notes by Paul Hacking) 9/8/09 () Physics (2) Quintic 3-fold (3) Math String theory is a N = 2 superconformal field theory (SCFT) which models elementary
More informationMirror Symmetry: Introduction to the B Model
Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds
More informationEnumerative Invariants in Algebraic Geometry and String Theory
Dan Abramovich -. Marcos Marino Michael Thaddeus Ravi Vakil Enumerative Invariants in Algebraic Geometry and String Theory Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 6-11,
More informationArithmetic Mirror Symmetry
Arithmetic Mirror Symmetry Daqing Wan April 15, 2005 Institute of Mathematics, Chinese Academy of Sciences, Beijing, P.R. China Department of Mathematics, University of California, Irvine, CA 92697-3875
More informationA note on quantum products of Schubert classes in a Grassmannian
J Algebr Comb (2007) 25:349 356 DOI 10.1007/s10801-006-0040-5 A note on quantum products of Schubert classes in a Grassmannian Dave Anderson Received: 22 August 2006 / Accepted: 14 September 2006 / Published
More informationarxiv: v1 [math.ag] 15 Jul 2018
. NOTE ON EQUIVARIANT I-FUNCTION OF LOCAL P n HYENHO LHO arxiv:1807.05501v1 [math.ag] 15 Jul 2018 Abstract. Several properties of a hyepergeometric series related to Gromov-Witten theory of some Calabi-Yau
More informationPERIODS OF STREBEL DIFFERENTIALS AND ALGEBRAIC CURVES DEFINED OVER THE FIELD OF ALGEBRAIC NUMBERS
PERIODS OF STREBEL DIFFERENTIALS AND ALGEBRAIC CURVES DEFINED OVER THE FIELD OF ALGEBRAIC NUMBERS MOTOHICO MULASE 1 AND MICHAEL PENKAVA 2 Abstract. In [8] we have shown that if a compact Riemann surface
More informationGromov-Witten invariants of hypersurfaces
Gromov-Witten invariants of hypersurfaces Dem Fachbereich Mathematik der Technischen Universität Kaiserslautern zur Erlangung der venia legendi vorgelegte Habilitationsschrift von Andreas Gathmann geboren
More informationRational curves on general type hypersurfaces
Rational curves on general type hypersurfaces Eric Riedl and David Yang May 5, 01 Abstract We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an
More informationAsymptotic of Enumerative Invariants in CP 2
Peking Mathematical Journal https://doi.org/.7/s4543-8-4-4 ORIGINAL ARTICLE Asymptotic of Enumerative Invariants in CP Gang Tian Dongyi Wei Received: 8 March 8 / Revised: 3 July 8 / Accepted: 5 July 8
More informationSemistability of certain bundles on a quintic Calabi-Yau threefold.
Semistability of certain bundles on a quintic Calabi-Yau threefold. Maria Chiara Brambilla Dipartimento di Matematica e Applicazioni per l Architettura, Università di Firenze, piazza Ghiberti, 27, 50122
More informationNovember In the course of another investigation we came across a sequence of polynomials
ON CERTAIN PLANE CURVES WITH MANY INTEGRAL POINTS F. Rodríguez Villegas and J. F. Voloch November 1997 0. In the course of another investigation we came across a sequence of polynomials P d Z[x, y], such
More informationIntersections in genus 3 and the Boussinesq hierarchy
ISSN: 1401-5617 Intersections in genus 3 and the Boussinesq hierarchy S. V. Shadrin Research Reports in Mathematics Number 11, 2003 Department of Mathematics Stockholm University Electronic versions of
More informationList of Works. Alessio Corti. (Last modified: March 2016)
List of Works Alessio Corti (Last modified: March 2016) Recent arxiv articles 1. The Sarkisov program for Mori fibred Calabi Yau pairs (with A.-S. Kaloghiros), arxiv:1504.00557, 12 pp., accepted for publication
More informationReferences Cited. [4] A. Geraschenko, S. Morrison, and R. Vakil, Mathoverflow, Notices of the Amer. Math. Soc. 57 no. 6, June/July 2010, 701.
References Cited [1] S. Billey and R. Vakil, Intersections of Schubert varieties and other permutation array schemes, in Algorithms in Algebraic Geometry, IMA Volume 146, p. 21 54, 2008. [2] I. Coskun
More informationGenerating functions and enumerative geometry
Generating functions and enumerative geometry Ragni Piene Centre of Mathematics for Applications and Department of Mathematics, University of Oslo October 26, 2005 KTH, Stockholm Generating functions Enumerative
More informationQuantum Cohomology Rings Generated by Gromov-Witten Invariants
REU Program 2017 Quantum Cohomology Rings Generated by Gromov-Witten Invariants By Nalinpat Ponoi Mentor : Prof. Anders Skovsted Buch Supported by SAST-ATPAC Backgrounds and Terminology What is Grassmannain?
More informationMathematical Results Inspired by Physics
ICM 2002 Vol. III 1 3 Mathematical Results Inspired by Physics Kefeng Liu Abstract I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems
More informationGauged Linear Sigma Model in the Geometric Phase
Gauged Linear Sigma Model in the Geometric Phase Guangbo Xu joint work with Gang Tian Princeton University International Conference on Differential Geometry An Event In Honour of Professor Gang Tian s
More informationString Duality and Moduli Spaces
String Duality and Moduli Spaces Kefeng Liu July 28, 2007 CMS and UCLA String Theory, as the unified theory of all fundamental forces, should be unique. But now there are Five different looking string
More informationOn the BCOV Conjecture
Department of Mathematics University of California, Irvine December 14, 2007 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called
More informationBlack Holes and Hurwitz Class Numbers
Black Holes and Hurwitz Class Numbers Shamit Kachru a,1, Arnav Tripathy b a Stanford Institute for Theoretical Physics Stanford University, Palo Alto, CA 94305, USA Email: skachru@stanford.edu b Department
More informationOn the tautological ring of M g,n
Proceedings of 7 th Gökova Geometry-Topology Conference, pp, 1 7 On the tautological ring of M g,n Tom Graber and Ravi Vakil 1. Introduction The purpose of this note is to prove: Theorem 1.1. R 0 (M g,n
More informationAlgebraic and Analytic Geometry WIT A Structured and Comprehensive Package for Computations in Intersection Theory
AMS SPECIAL SESSION Computational Algebraic and Analytic Geometry for Low dimensional Varieties WIT A Structured and Comprehensive Package for Computations in Intersection Theory SEBASTIAN XAMBÓ DESCAMPS
More informationOn the WDVV-equation in quantum K-theory
On the WDVV-equation in quantum K-theory Alexander Givental UC Berkeley and Caltech 0. Introduction. Quantum cohomology theory can be described in general words as intersection theory in spaces of holomorphic
More informationTAUTOLOGICAL EQUATION IN M 3,1 VIA INVARIANCE THEOREM
TAUTOLOGICAL EQUATION IN M 3, VIA INVARIANCE THEOREM D. ARCARA AND Y.-P. LEE This paper is dedicated to Matteo S. Arcara, who was born while the paper was at its last phase of preparation. Abstract. A
More informationTHE EFFECTIVE CONE OF THE KONTSEVICH MODULI SPACE
THE EFFECTIVE CONE OF THE KONTSEVICH MODULI SPACE IZZET COSKUN, JOE HARRIS, AND JASON STARR Abstract. In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable
More informationTopics in Geometry: Mirror Symmetry
MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:
More informationCANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES
CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES JULIUS ROSS This short survey aims to introduce some of the ideas and conjectures relating stability of projective varieties to the existence of
More informationPseudoholomorphic Curves and Mirror Symmetry
Pseudoholomorphic Curves and Mirror Symmetry Santiago Canez February 14, 2006 Abstract This survey article was written for Prof. Alan Weinstein s Symplectic Geometry (Math 242) course at UC Berkeley in
More informationarxiv: v2 [math.ag] 3 Mar 2008 Victor BATYREV *
Constructing new Calabi Yau 3-folds and their mirrors via conifold transitions arxiv:0802.3376v2 [math.ag] 3 Mar 2008 Victor BATYREV * Mathematisches Institut, Universität Tübingen Auf der Morgenstelle
More informationA POLAR, THE CLASS AND PLANE JACOBIAN CONJECTURE
Bull. Korean Math. Soc. 47 (2010), No. 1, pp. 211 219 DOI 10.4134/BKMS.2010.47.1.211 A POLAR, THE CLASS AND PLANE JACOBIAN CONJECTURE Dosang Joe Abstract. Let P be a Jacobian polynomial such as deg P =
More informationTopological Strings and Donaldson-Thomas invariants
Topological Strings and Donaldson-Thomas invariants University of Patras Πανɛπιστήµιo Πατρών RTN07 Valencia - Short Presentation work in progress with A. Sinkovics and R.J. Szabo Topological Strings on
More informationThen the blow up of V along a line is a rational conic bundle over P 2. Definition A k-rational point of a scheme X over S is any point which
16. Cubics II It turns out that the question of which varieties are rational is one of the subtlest geometric problems one can ask. Since the problem of determining whether a variety is rational or not
More informationTHE JACOBIAN AND FORMAL GROUP OF A CURVE OF GENUS 2 OVER AN ARBITRARY GROUND FIELD E. V. Flynn, Mathematical Institute, University of Oxford
THE JACOBIAN AND FORMAL GROUP OF A CURVE OF GENUS 2 OVER AN ARBITRARY GROUND FIELD E. V. Flynn, Mathematical Institute, University of Oxford 0. Introduction The ability to perform practical computations
More informationA tourist s guide to intersection theory on moduli spaces of curves
A tourist s guide to intersection theory on moduli spaces of curves The University of Melbourne In the past few decades, moduli spaces of curves have attained notoriety amongst mathematicians for their
More informationA RECONSTRUCTION THEOREM IN QUANTUM COHOMOLOGY AND QUANTUM K-THEORY
A RECONSTRUCTION THEOREM IN QUANTUM COHOMOLOGY AND QUANTUM K-THEORY Y.-P. LEE AND R. PANDHARIPANDE Abstract. A reconstruction theorem for genus 0 gravitational quantum cohomology and quantum K-theory is
More informationGenerating functions in enumerative geometry
Generating functions in enumerative geometry Ragni Piene ICREM 5 Bandung, Indonesia October 22, 2011 Counting 1 Counting 2 Counting 3 Partitions Let n be a positive integer. In how many ways p(n) can we
More informationOn Lines and Joints. August 17, Abstract
On Lines and Joints Haim Kaplan Micha Sharir Eugenii Shustin August 17, 2009 Abstract Let L be a set of n lines in R d, for d 3. A joint of L is a point incident to at least d lines of L, not all in a
More informationIntersection theory on moduli spaces of curves via hyperbolic geometry
Intersection theory on moduli spaces of curves via hyperbolic geometry The University of Melbourne In the past few decades, moduli spaces of curves have become the centre of a rich confluence of rather
More informationarxiv: v1 [math.ag] 28 Sep 2016
LEFSCHETZ CLASSES ON PROJECTIVE VARIETIES JUNE HUH AND BOTONG WANG arxiv:1609.08808v1 [math.ag] 28 Sep 2016 ABSTRACT. The Lefschetz algebra L X of a smooth complex projective variety X is the subalgebra
More informationModuli of Pointed Curves. G. Casnati, C. Fontanari
Moduli of Pointed Curves G. Casnati, C. Fontanari 1 1. Notation C is the field of complex numbers, GL k the general linear group of k k matrices with entries in C, P GL k the projective linear group, i.e.
More informationTHE CAPORASO-HARRIS FORMULA AND PLANE RELATIVE GROMOV-WITTEN INVARIANTS IN TROPICAL GEOMETRY
THE CAPORASO-HARRIS FORMULA AND PLANE RELATIVE GROMOV-WITTEN INVARIANTS IN TROPICAL GEOMETRY ANDREAS GATHMANN AND HANNAH MARKWIG Abstract. Some years ago Caporaso and Harris have found a nice way to compute
More informationKnot Contact Homology, Chern-Simons Theory, and Topological Strings
Knot Contact Homology, Chern-Simons Theory, and Topological Strings Tobias Ekholm Uppsala University and Institute Mittag-Leffler, Sweden Symplectic Topology, Oxford, Fri Oct 3, 2014 Plan Reports on joint
More informationStatistical Mechanics & Enumerative Geometry:
Statistical Mechanics & Enumerative Geometry: Christian Korff (ckorff@mathsglaacuk) University Research Fellow of the Royal Society Department of Mathematics, University of Glasgow joint work with C Stroppel
More informationCubic hypersurfaces over finite fields
Cubic hypersurfaces over finite fields Olivier DEBARRE (joint with Antonio LAFACE and Xavier ROULLEAU) École normale supérieure, Paris, France Moscow, HSE, September 2015 Olivier DEBARRE Cubic hypersurfaces
More informationGeneralized Tian-Todorov theorems
Generalized Tian-Todorov theorems M.Kontsevich 1 The classical Tian-Todorov theorem Recall the classical Tian-Todorov theorem (see [4],[5]) about the smoothness of the moduli spaces of Calabi-Yau manifolds:
More informationGeometry of the Calabi-Yau Moduli
Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51
More informationQUALITATIVE ASPECTS OF COUNTING REAL RATIONAL CURVES ON REAL K3 SURFACES
QUALITATIVE ASPECTS OF COUNTING REAL RATIONAL CURVES ON REAL K3 SURFACES VIATCHESLAV KHARLAMOV AND RAREŞ RĂSDEACONU 1 ABSTRACT. We study qualitative aspects of the Welschinger-like Z-valued count of real
More informationGromov Witten Theory of Blowups of Toric Threefolds
Gromov Witten Theory of Blowups of Toric Threefolds Dhruv Ranganathan Dagan Karp, Advisor Davesh Maulik, Reader May, 2012 Department of Mathematics Copyright c 2012 Dhruv Ranganathan. The author grants
More informationPublication. * are expository articles.
Publication * are expository articles. [1] A finiteness theorem for negatively curved manifolds, J. Differential Geom. 20 (1984) 497-521. [2] Theory of Convergence for Riemannian orbifolds, Japanese J.
More informationEXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION
EXAMPLES OF CALABI-YAU 3-MANIFOLDS WITH COMPLEX MULTIPLICATION JAN CHRISTIAN ROHDE Introduction By string theoretical considerations one is interested in Calabi-Yau manifolds since Calabi-Yau 3-manifolds
More informationThe Number of Rational Points on Elliptic Curves and Circles over Finite Fields
Vol:, No:7, 008 The Number of Rational Points on Elliptic Curves and Circles over Finite Fields Betül Gezer, Ahmet Tekcan, and Osman Bizim International Science Index, Mathematical and Computational Sciences
More informationPorteous s Formula for Maps between Coherent Sheaves
Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for
More informationThe geometry of Landau-Ginzburg models
Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror
More informationCURRICULUM VITAE. Di Yang. Date of Birth: Feb. 8, 1986 Nationality: P. R. China
CURRICULUM VITAE Di Yang PERSONAL INFORMATION Name: Di Family Name: Yang Date of Birth: Feb. 8, 1986 Nationality: P. R. China Address: MPIM, Vivatsgasse 7, Bonn 53111, Germany Phone Number: +49-15163351572
More informationVARIETIES WITHOUT EXTRA AUTOMORPHISMS I: CURVES BJORN POONEN
VARIETIES WITHOUT EXTRA AUTOMORPHISMS I: CURVES BJORN POONEN Abstract. For any field k and integer g 3, we exhibit a curve X over k of genus g such that X has no non-trivial automorphisms over k. 1. Statement
More informationConstructing new Calabi Yau 3-folds and their mirrors via conifold transitions
c 2010 International Press Adv. Theor. Math. Phys. 14 (2010) 879 898 Constructing new Calabi Yau 3-folds and their mirrors via conifold transitions Victor Batyrev 1 and Maximilian Kreuzer 2 1 Mathematisches
More informationALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3
ALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3 IZZET COSKUN AND ERIC RIEDL Abstract. We prove that a curve of degree dk on a very general surface of degree d 5 in P 3 has geometric
More informationString-Theory: Open-closed String Moduli Spaces
String-Theory: Open-closed String Moduli Spaces Heidelberg, 13.10.2014 History of the Universe particular: Epoch of cosmic inflation in the early Universe Inflation and Inflaton φ, potential V (φ) Possible
More informationA Partial Multivariate Polynomial Interpolation Problem
International Mathematical Forum, 2, 2007, no. 62, 3089-3094 A Partial Multivariate Polynomial Interpolation Problem E. Ballico 1 Dept. of Mathematics University of Trento 38050 Povo (TN), Italy ballico@science.unitn.it
More informationA WHIRLWIND TOUR BEYOND QUADRATICS Steven J. Wilson, JCCC Professor of Mathematics KAMATYC, Wichita, March 4, 2017
b x1 u v a 9abc b 7a d 7a d b c 4ac 4b d 18abcd u 4 b 1 i 1 i 54a 108a x u v where a 9abc b 7a d 7a d b c 4ac 4b d 18abcd v 4 b 1 i 1 i 54a x u v 108a a //017 A WHIRLWIND TOUR BEYOND QUADRATICS Steven
More informationThe problematic art of counting
The problematic art of counting Ragni Piene SMC Stockholm November 16, 2011 Mikael Passare A (very) short history of counting: tokens and so on... Partitions Let n be a positive integer. In how many ways
More informationF -SINGULARITIES AND FROBENIUS SPLITTING NOTES 12/9-2010
F -SINGULARITIES AND FROBENIUS SPLITTING NOTES 12/9-2010 KARL SCHWEDE 1. Fujita s conjecture We begin with a discussion of Castlenuovo-regularity, see [Laz04, Section 1.8]. Definition 1.1. Let F be a coherent
More informationCounting surfaces of any topology, with Topological Recursion
Counting surfaces of any topology, with Topological Recursion 1... g 2... 3 n+1 = 1 g h h I 1 + J/I g 1 2 n+1 Quatum Gravity, Orsay, March 2013 Contents Outline 1. Introduction counting surfaces, discrete
More informationINTERSECTION-THEORETICAL COMPUTATIONS ON M g
PARAMETER SPACES BANACH CENTER PUBLICATIONS, VOLUME 36 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1996 INTERSECTION-THEORETICAL COMPUTATIONS ON M g CAREL FABE R Faculteit der Wiskunde
More informationStationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture
Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture Alexander Givental UC Berkeley July 8, 1996, Revised December 1, 1996 0. Introduction. Consider the differential
More informationHow many elliptic curves can have the same prime conductor? Alberta Number Theory Days, BIRS, 11 May Noam D. Elkies, Harvard University
How many elliptic curves can have the same prime conductor? Alberta Number Theory Days, BIRS, 11 May 2013 Noam D. Elkies, Harvard University Review: Discriminant and conductor of an elliptic curve Finiteness
More informationTopological Classification of Morse Functions and Generalisations of Hilbert s 16-th Problem
Math Phys Anal Geom (2007) 10:227 236 DOI 10.1007/s11040-007-9029-0 Topological Classification of Morse Functions and Generalisations of Hilbert s 16-th Problem Vladimir I. Arnold Received: 30 August 2007
More informationA TOOL FOR STABLE REDUCTION OF CURVES ON SURFACES
A TOOL FOR STABLE REDUCTION OF CURVES ON SURFACES RAVI VAKIL Abstract. In the study of the geometry of curves on surfaces, the following question often arises: given a one-parameter family of divisors
More informationSPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS
SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree
More informationQuantum cohomology of homogeneous varieties: a survey Harry Tamvakis
Quantum cohomology of homogeneous varieties: a survey Harry Tamvakis Let G be a semisimple complex algebraic group and P a parabolic subgroup of G The homogeneous space X = G/P is a projective complex
More informationBasic Algebraic Geometry 1
Igor R. Shafarevich Basic Algebraic Geometry 1 Second, Revised and Expanded Edition Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Table of Contents Volume 1
More informationAbstracts of papers. Amod Agashe
Abstracts of papers Amod Agashe In this document, I have assembled the abstracts of my work so far. All of the papers mentioned below are available at http://www.math.fsu.edu/~agashe/math.html 1) On invisible
More informationIAN STRACHAN. 1. Introduction
HOW TO COUNT CURVES: FROM 19 th CENTURY PROBLEMS TO 21 st CENTURY SOLUTIONS IAN STRACHAN Guess the next term in the sequence: 1. Introduction 1, 1, 12, 620, 87304, 26312976. This problem belongs to an
More informationREPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES. Ken Ono. Dedicated to the memory of Robert Rankin.
REPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES Ken Ono Dedicated to the memory of Robert Rankin.. Introduction and Statement of Results. If s is a positive integer, then let rs; n denote the number of
More informationRemarks on Chern-Simons Theory. Dan Freed University of Texas at Austin
Remarks on Chern-Simons Theory Dan Freed University of Texas at Austin 1 MSRI: 1982 2 Classical Chern-Simons 3 Quantum Chern-Simons Witten (1989): Integrate over space of connections obtain a topological
More informationOn the Length of Lemniscates
On the Length of Lemniscates Alexandre Eremenko & Walter Hayman For a monic polynomial p of degree d, we write E(p) := {z : p(z) =1}. A conjecture of Erdős, Herzog and Piranian [4], repeated by Erdős in
More informationOn Arithmetic Properties of Bell Numbers, Delannoy Numbers and Schröder Numbers
A tal given at the Institute of Mathematics, Academia Sinica (Taiwan (Taipei; July 6, 2011 On Arithmetic Properties of Bell Numbers, Delannoy Numbers and Schröder Numbers Zhi-Wei Sun Nanjing University
More informationRANKIN-COHEN BRACKETS AND VAN DER POL-TYPE IDENTITIES FOR THE RAMANUJAN S TAU FUNCTION
RANKIN-COHEN BRACKETS AND VAN DER POL-TYPE IDENTITIES FOR THE RAMANUJAN S TAU FUNCTION B. RAMAKRISHNAN AND BRUNDABAN SAHU Abstract. We use Rankin-Cohen brackets for modular forms and quasimodular forms
More informationLocal and log BPS invariants
Local and log BPS invariants Jinwon Choi and Michel van Garrel joint with S. Katz and N. Takahashi Sookmyung Women s University KIAS August 11, 2016 J. Choi (Sookmyung) Local and log BPS invariants Aug
More informationCoordinate Change of Gauss-Manin System and Generalized Mirror Transformation arxiv:math/ v2 [math.ag] 12 Nov 2003
Coordinate Change of Gauss-Manin System and Generalized Mirror Transformation arxiv:math/030v [math.ag] Nov 003 Masao Jinzenji Division of Mathematics, Graduate School of Science Hokkaido University Kita-ku,
More informationOn the zeros of certain modular forms
On the zeros of certain modular forms Masanobu Kaneko Dedicated to Professor Yasutaka Ihara on the occasion of his 60th birthday. The aim of this short note is to list several families of modular forms
More informationFOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2
FOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2 CAMERON FRANC AND GEOFFREY MASON Abstract. We prove the following Theorem. Suppose that F = (f 1, f 2 ) is a 2-dimensional vector-valued
More informationGAUGED LINEAR SIGMA MODEL SPACES
GAUGED LINEAR SIGMA MODEL SPACES FELIPE CASTELLANO-MACIAS ADVISOR: FELIX JANDA Abstract. The gauged linear sigma model (GLSM) originated in physics but it has recently made it into mathematics as an enumerative
More informationEisenstein type series for Calabi-Yau varieties 1
Eisenstein type series for Calabi-Yau varieties Hossein Movasati Instituto de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina,, 226-32, Rio de Janeiro, RJ, Brazil, www.impa.br/ hossein, hossein@impa.br
More informationCOMPOSITIONS, PARTITIONS, AND FIBONACCI NUMBERS
COMPOSITIONS PARTITIONS AND FIBONACCI NUMBERS ANDREW V. SILLS Abstract. A bijective proof is given for the following theorem: the number of compositions of n into odd parts equals the number of compositions
More information