On the Use of Parameter and Moduli Spaces in Curve Counting
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Schubert Calculus According to Schubert
Schubert Calculus according to Schubert Felice Ronga February 16, 2006 Abstract We try to understand and justify Schubert calculus the way Schubert did it. 1 Introduction In his famous book [7] “Kalk¨ulder abz¨ahlende Geometrie”, published in 1879, Dr. Hermann C. H. Schubert has developed a method for solving problems of enumerative geometry, called Schubert Calculus today, and has applied it to a great number of cases. This book is self-contained : given some aptitude to the mathematical reasoning, a little geometric intuition and a good knowledge of the german language, one can enjoy the many enumerative problems that are presented and solved. Hilbert’s 15th problems asks to give a rigourous foundation to Schubert’s method. This has been largely accomplished using intersection theory (see [4],[5], [2]), and most of Schubert’s calculations have been con- firmed. Our purpose is to understand and justify the very method that Schubert has used. We will also step through his calculations in some simple cases, in order to illustrate Schubert’s way of proceeding. Here is roughly in what Schubert’s method consists. First of all, we distinguish basic elements in the complex projective space : points, planes, lines. We shall represent by symbols, say x, y, conditions (in german : Bedingungen) that some geometric objects have to satisfy; the product x · y of two conditions represents the condition that x and y are satisfied, the sum x + y represents the condition that x or y is satisfied. The conditions on the basic elements that can be expressed using other basic elements (for example : the lines in space that must go through a given point) satisfy a number of formulas that can be determined rather easily by geometric reasoning. -
Bézout's Theorem
Bézout's theorem Toni Annala Contents 1 Introduction 2 2 Bézout's theorem 4 2.1 Ane plane curves . .4 2.2 Projective plane curves . .6 2.3 An elementary proof for the upper bound . 10 2.4 Intersection numbers . 12 2.5 Proof of Bézout's theorem . 16 3 Intersections in Pn 20 3.1 The Hilbert series and the Hilbert polynomial . 20 3.2 A brief overview of the modern theory . 24 3.3 Intersections of hypersurfaces in Pn ...................... 26 3.4 Intersections of subvarieties in Pn ....................... 32 3.5 Serre's Tor-formula . 36 4 Appendix 42 4.1 Homogeneous Noether normalization . 42 4.2 Primes associated to a graded module . 43 1 Chapter 1 Introduction The topic of this thesis is Bézout's theorem. The classical version considers the number of points in the intersection of two algebraic plane curves. It states that if we have two algebraic plane curves, dened over an algebraically closed eld and cut out by multi- variate polynomials of degrees n and m, then the number of points where these curves intersect is exactly nm if we count multiple intersections and intersections at innity. In Chapter 2 we dene the necessary terminology to state this theorem rigorously, give an elementary proof for the upper bound using resultants, and later prove the full Bézout's theorem. A very modest background should suce for understanding this chapter, for example the course Algebra II given at the University of Helsinki should cover most, if not all, of the prerequisites. In Chapter 3, we generalize Bézout's theorem to higher dimensional projective spaces. -
The Historical Development of Algebraic Geometry∗
The Historical Development of Algebraic Geometry∗ Jean Dieudonn´e March 3, 1972y 1 The lecture This is an enriched transcription of footage posted by the University of Wis- consin{Milwaukee Department of Mathematical Sciences [1]. The images are composites of screenshots from the footage manipulated with Python and the Python OpenCV library [2]. These materials were prepared by Ryan C. Schwiebert with the goal of capturing and restoring some of the material. The lecture presents the content of Dieudonn´e'sarticle of the same title [3] as well as the content of earlier lecture notes [4], but the live lecture is natu- rally embellished with different details. The text presented here is, for the most part, written as it was spoken. This is why there are some ungrammatical con- structions of spoken word, and some linguistic quirks of a French speaker. The transcriber has taken some liberties by abbreviating places where Dieudonn´e self-corrected. That is, short phrases which turned out to be false starts have been elided, and now only the subsequent word-choices that replaced them ap- pear. Unfortunately, time has taken its toll on the footage, and it appears that a brief portion at the beginning, as well as the last enumerated period (VII. Sheaves and schemes), have been lost. (Fortunately, we can still read about them in [3]!) The audio and video quality has contributed to several transcription errors. Finally, a lack of thorough knowledge of the history of algebraic geometry has also contributed some errors. Contact with suggestions for improvement of arXiv:1803.00163v1 [math.HO] 1 Mar 2018 the materials is welcome. -
INTERSECTION NUMBERS, TRANSFERS, and GROUP ACTIONS by Daniel H. Gottlieb and Murad Ozaydin Purdue University University of Oklah
INTERSECTION NUMBERS, TRANSFERS, AND GROUP ACTIONS by Daniel H. Gottlieb and Murad Ozaydin Purdue University University of Oklahoma Abstract We introduce a new definition of intersection number by means of umkehr maps. This definition agrees with an obvious definition up to sign. It allows us to extend the concept parametrically to fibre bundles. For fibre bundles we can then define intersection number transfers. Fibre bundles arise naturally in equivariant situations, so the trace of the action divides intersection numbers. We apply this to equivariant projective varieties and other examples. Also we study actions of non-connected groups and their traces. AMS subject classification 1991: Primary: 55R12, 57S99, 55M25 Secondary: 14L30 Keywords: Poincare Duality, trace of action, Sylow subgroups, Euler class, projective variety. 1 1. Introduction When two manifolds with complementary dimensions inside a third manifold intersect transversally, their intersection number is defined. This classical topological invariant has played an important role in topology and its applications. In this paper we introduce a new definition of intersection number. The definition naturally extends to the case of parametrized manifolds over a base B; that is to fibre bundles over B . Then we can define transfers for the fibre bundles related to the intersection number. This is our main objective , and it is done in Theorem 6. The method of producing transfers depends upon the Key Lemma which we prove here. Special cases of this lemma played similar roles in the establishment of the Euler–Poincare transfer, the Lefschetz number transfer, and the Nakaoka transfer. Two other features of our definition are: 1) We replace the idea of submanifolds with the idea of maps from manifolds, and in effect we are defining the intersection numbers of the maps; 2) We are not restricted to only two maps, we can define the intersection number for several maps, or submanifolds. -
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. -
Enumerative Geometry of Rational Space Curves
ENUMERATIVE GEOMETRY OF RATIONAL SPACE CURVES DANIEL F. CORAY [Received 5 November 1981] Contents 1. Introduction A. Rational space curves through a given set of points 263 B. Curves on quadrics 265 2. The two modes of degeneration A. Incidence correspondences 267 B. Preliminary study of the degenerations 270 3. Multiplicities A. Transversality of the fibres 273 B. A conjecture 278 4. Determination of A2,3 A. The Jacobian curve of the associated net 282 B. The curve of moduli 283 References 287 1. Introduction A. Rational space curves through a given set of points It is a classical result of enumerative geometry that there is precisely one twisted cubic through a set of six points in general position in P3. Several proofs are available in the literature; see, for example [15, §11, Exercise 4; or 12, p. 186, Example 11]. The present paper is devoted to the generalization of this result to rational curves of higher degrees. < Let m be a positive integer; we denote by €m the Chow variety of all curves with ( degree m in PQ. As is well-known (cf. Lemma 2.4), €m has an irreducible component 0im, of dimension Am, whose general element is the Chow point of a smooth rational curve. Moreover, the Chow point of any irreducible space curve with degree m and geometric genus zero belongs to 0tm. Except for m = 1 or 2, very little is known on the geometry of this variety. So we may also say that the object of this paper is to undertake an enumerative study of the variety $%m. -
3264 Conics in a Second
3264 Conics in a Second Paul Breiding, Bernd Sturmfels, and Sascha Timme This article and its accompanying web interface present infinity, provided 퐴 and 푈 are irreducible and not multi- Steiner’s conic problem and a discussion on how enumerative ples of each other. This is the content of B´ezout’s theorem. and numerical algebraic geometry complement each other. To take into account the points of intersection at infinity, The intended audience is students at an advanced under- algebraic geometers like to replace the affine plane ℂ2 with 2 grad level. Our readers can see current computational the complex projective plane ℙℂ. In the following, when tools in action on a geometry problem that has inspired we write “count,” we always mean counting solutions in scholars for two centuries. The take-home message is that projective space. Nevertheless, for our exposition we work numerical methods in algebraic geometry are fast and reli- with ℂ2. able. A solution (푥, 푦) of the system 퐴 = 푈 = 0 has multiplic- We begin by recalling the statement of Steiner’s conic ity ≥ 2 if it is a zero of the Jacobian determinant 2 problem. A conic in the plane ℝ is the set of solutions to 휕퐴 휕푈 휕퐴 휕푈 2 ⋅ − ⋅ = 2(푎1푢2 − 푎2푢1)푥 a quadratic equation 퐴(푥, 푦) = 0, where 휕푥 휕푦 휕푦 휕푥 (3) 2 2 +4(푎1푢3 − 푎3푢1)푥푦 + ⋯ + (푎4푢5 − 푎5푢4). 퐴(푥, 푦) = 푎1푥 + 푎2푥푦 + 푎3푦 + 푎4푥 + 푎5푦 + 푎6. (1) Geometrically, the conic 푈 is tangent to the conic 퐴 if (1), If there is a second conic (2), and (3) are zero for some (푥, 푦) ∈ ℂ2. -
Arxiv:1710.06183V2 [Math.AG]
ALGEBRAIC INTEGRABILITY OF FOLIATIONS WITH NUMERICALLY TRIVIAL CANONICAL BUNDLE ANDREAS HÖRING AND THOMAS PETERNELL Abstract. Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent work of Druel and Greb-Guenancia-Kebekus this establishes the Beauville-Bogomolov decomposition for minimal models with trivial canon- ical class. 1. introduction 1.A. Main result. Let X be a normal complex projective variety that is smooth in codimension two, and let E be a reflexive sheaf on X. If E is slope-stable with respect to some ample divisor H of slope µH (E)=0, then a famous result of Mehta-Ramanathan [MR84] says that the restriction EC to a general complete intersection C of sufficiently ample divisors is stable and nef. On the other hand the variety X contains many dominating families of irreducible curves to which Mehta-Ramanathan does not apply; therefore one expects that, apart from very special situations, EC will not be nef for many curves C. If E is locally free, denote by π : P(E) → X the projectivisation of E and by ζ := c1(OP(E)(1)) the tautological class on P(E). The nefness of EC then translates into the nefness of the restriction of ζ to P(EC). Thus, the stability of E implies some positivity of the tautological class ζ. On the other hand, the non-nefness of EC on many curves can be rephrased by saying that the tautological class ζ should not be pseudoeffective. -
Enumerative Geometry
Lecture 1 Renzo Cavalieri Enumerative Geometry Enumerative geometry is an ancient branch of mathematics that is concerned with counting geometric objects that satisfy a certain number of geometric con- ditions. Here are a few examples of typical enumerative geometry questions: Q1: How many lines pass through 2 points in the plane? Q2: How many conics pass through 5 points in the plane? 1 Q3: How many rational cubics (i.e. having one node) pass through 8 points in the plane? Qd: How many rational curves of degree d pass through 3d − 1 points in the plane? OK, well, these are all part of one big family...here is one of a slightly different flavor: Ql: How many lines pass through 4 lines in three dimensional space? Some Observations: 1. I’ve deliberately left somewhat vague what the ambient space of our geo- metric objects: for one, I don’t want to worry too much about it; second, if you like, for example, to work over funky number fields, then by all means these can still be interesting questions. In order to get nice answers we will be working over the complex numbers (where we have the fundamental theorem of algebra working for us). Also, when most algebraic geometers say things like “plane”, what they really mean is an appropriate com- pactification of it...there’s a lot of reasons to prefer working on compact spaces...but this is a slightly different story... 2. You might complain that the questions may have different answers, be- cause I said nothing about how the points are distributed on the plane. -
Intersection Theory
APPENDIX A Intersection Theory In this appendix we will outline the generalization of intersection theory and the Riemann-Roch theorem to nonsingular projective varieties of any dimension. To motivate the discussion, let us look at the case of curves and surfaces, and then see what needs to be generalized. For a divisor D on a curve X, leaving out the contribution of Serre duality, we can write the Riemann-Roch theorem (IV, 1.3) as x(.!Z'(D)) = deg D + 1 - g, where xis the Euler characteristic (III, Ex. 5.1). On a surface, we can write the Riemann-Roch theorem (V, 1.6) as 1 x(!l'(D)) = 2 D.(D - K) + 1 + Pa· In each case, on the left-hand side we have something involving cohomol ogy groups of the sheaf !l'(D), while on the right-hand side we have some numerical data involving the divisor D, the canonical divisor K, and some invariants of the variety X. Of course the ultimate aim of a Riemann-Roch type theorem is to compute the dimension of the linear system IDI or of lnDI for large n (II, Ex. 7.6). This is achieved by combining a formula for x(!l'(D)) with some vanishing theorems for Hi(X,!l'(D)) fori > 0, such as the theorems of Serre (III, 5.2) or Kodaira (III, 7.15). We will now generalize these results so as to give an expression for x(!l'(D)) on a nonsingular projective variety X of any dimension. And while we are at it, with no extra effort we get a formula for x(t&"), where @" is any coherent locally free sheaf. -
Enumerative Geometry and Geometric Representation Theory
Enumerative geometry and geometric representation theory Andrei Okounkov Abstract This is an introduction to: (1) the enumerative geometry of rational curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer Institute. 1 Introduction 1.1 These notes are written to accompany my lectures given in Salt Lake City in 2015. With modern technology, one should be able to access the materials from those lectures from anywhere in the world, so a text transcribing them is not really needed. Instead, I will try to spend more time on points that perhaps require too much notation for a broadly aimed series of talks, but which should ease the transition to reading detailed lecture notes like [90]. The fields from the title are vast and they intersect in many different ways. Here we will talk about a particular meeting point, the progress at which in the time since Seattle I find exciting enough to report in Salt Lake City. The advances both in the subject matter itself, and in my personal understanding of it, owe a lot to M. Aganagic, R. Bezrukavnikov, P. Etingof, D. Maulik, N. Nekrasov, and others, as will be clear from the narrative. 1.2 The basic question in representation theory is to describe the homomorphisms arXiv:1701.00713v1 [math.AG] 3 Jan 2017 some algebra A matrices , (1) Ñ and the geometric representation theory aims to describe the source, the target, the map itself, or all of the above, geometrically. For example, matrices may be replaced by correspondences, by which we mean cycles in X X, where X is an algebraic variety, or K-theory classes on X X et cetera. -
An Enumerative Geometry Framework for Algorithmic Line Problems in R3∗
SIAM J. COMPUT. c 2002 Society for Industrial and Applied Mathematics Vol. 31, No. 4, pp. 1212–1228 AN ENUMERATIVE GEOMETRY FRAMEWORK FOR ALGORITHMIC LINE PROBLEMS IN R3∗ THORSTEN THEOBALD† Abstract. We investigate the enumerative geometry aspects of algorithmic line problems when the admissible bodies are balls or polytopes. For this purpose, we study the common tangent lines/transversals to k balls of arbitrary radii and 4 − k lines in R3. In particular, we compute tight upper bounds for the maximum number of real common tangents/transversals in these cases. Our results extend the results of Macdonald, Pach, and Theobald who investigated common tangents to four unit balls in R3 [Discrete Comput. Geom., 26 (2001), pp. 1–17]. Key words. tangents, balls, transversals, lines, enumerative geometry, real solutions, computa- tional geometry AMS subject classifications. 14N10, 68U05, 51M30, 14P99, 52C45 PII. S009753970038050X 1. Introduction. Algorithmic questions involving lines in R3 belong to the fun- damental problems in computational geometry [36, 26], computer graphics [28], and robotics [33]. As an initial reference example from computational geometry, consider the problem of determining which bodies of a given scene cannot be seen from any viewpoint outside of the scene. From the geometric point of view, this leads to the problem of determining the common tangents to four given bodies in R3 (cf. section 2). Other algorithmic tasks leading to the same geometric core problem include com- puting smallest enclosing cylinders [32], computing geometric permutations/stabbing lines [27, 2], controlling a laser beam in manufacturing [26], or solving placement problems in geometric modeling [10, 17].