Singularities of Integrable Systems and Nodal Curves
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Arxiv:1810.08742V1 [Math.CV] 20 Oct 2018 Centroid of the Points Zi and Ei = Zi − C
SOME REMARKS ON THE CORRESPONDENCE BETWEEN ELLIPTIC CURVES AND FOUR POINTS IN THE RIEMANN SPHERE JOSE´ JUAN-ZACAR´IAS Abstract. In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in the Hesse normal form. In this setting, we give an alternative proof of the equivalence betweeen the Edwards and the Jacobi normal forms. Also, we give a geometric construction of the cross ratios for 4-point sets in general position. Introduction A complex elliptic curve is by definition a compact Riemann surface of genus 1. By the uniformization theorem, every elliptic curve is conformally equivalent to an algebraic curve given by a cubic polynomial in the form 2 3 3 2 (1) E : y = 4x − g2x − g3; with ∆ = g2 − 27g3 6= 0; this is called the Weierstrass normal form. For computational or geometric pur- poses it may be necessary to find a Weierstrass normal form for an elliptic curve, which could have been given by another equation. At best, we could predict the right changes of variables in order to transform such equation into a Weierstrass normal form, but in general this is a difficult process. A different method to find the normal form (1) for a given elliptic curve, avoiding any change of variables, requires a degree 2 meromorphic function on the elliptic curve, which by a classical theorem always exists and in many cases it is not difficult to compute. -
Riemann Surfaces
RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them. -
Automorphisms of a Symmetric Product of a Curve (With an Appendix by Najmuddin Fakhruddin)
Documenta Math. 1181 Automorphisms of a Symmetric Product of a Curve (with an Appendix by Najmuddin Fakhruddin) Indranil Biswas and Tomas´ L. Gomez´ Received: February 27, 2016 Revised: April 26, 2017 Communicated by Ulf Rehmann Abstract. Let X be an irreducible smooth projective curve of genus g > 2 defined over an algebraically closed field of characteristic dif- ferent from two. We prove that the natural homomorphism from the automorphisms of X to the automorphisms of the symmetric product Symd(X) is an isomorphism if d > 2g − 2. In an appendix, Fakhrud- din proves that the isomorphism class of the symmetric product of a curve determines the isomorphism class of the curve. 2000 Mathematics Subject Classification: 14H40, 14J50 Keywords and Phrases: Symmetric product; automorphism; Torelli theorem. 1. Introduction Automorphisms of varieties is currently a very active topic in algebraic geom- etry; see [Og], [HT], [Zh] and references therein. Hurwitz’s automorphisms theorem, [Hu], says that the order of the automorphism group Aut(X) of a compact Riemann surface X of genus g ≥ 2 is bounded by 84(g − 1). The group of automorphisms of the Jacobian J(X) preserving the theta polariza- tion is generated by Aut(X), translations and inversion [We], [La]. There is a universal constant c such that the order of the group of all automorphisms of any smooth minimal complex projective surface S of general type is bounded 2 above by c · KS [Xi]. Let X be a smooth projective curve of genus g, with g > 2, over an al- gebraically closed field of characteristic different from two. -
Associating Abelian Varieties to Weight-2 Modular Forms: the Eichler-Shimura Construction
Ecole´ Polytechnique Fed´ erale´ de Lausanne Master’s Thesis in Mathematics Associating abelian varieties to weight-2 modular forms: the Eichler-Shimura construction Author: Corentin Perret-Gentil Supervisors: Prof. Akshay Venkatesh Stanford University Prof. Philippe Michel EPF Lausanne Spring 2014 Abstract This document is the final report for the author’s Master’s project, whose goal was to study the Eichler-Shimura construction associating abelian va- rieties to weight-2 modular forms for Γ0(N). The starting points and main resources were the survey article by Fred Diamond and John Im [DI95], the book by Goro Shimura [Shi71], and the book by Fred Diamond and Jerry Shurman [DS06]. The latter is a very good first reference about this sub- ject, but interesting points are sometimes eluded. In particular, although most statements are given in the general setting, the book mainly deals with the particular case of elliptic curves (i.e. with forms having rational Fourier coefficients), with little details about abelian varieties. On the other hand, Chapter 7 of Shimura’s book is difficult, according to the author himself, and the article by Diamond and Im skims rapidly through the subject, be- ing a survey. The goal of this document is therefore to give an account of the theory with intermediate difficulty, accessible to someone having read a first text on modular forms – such as [Zag08] – and with basic knowledge in the theory of compact Riemann surfaces (see e.g. [Mir95]) and algebraic geometry (see e.g. [Har77]). This report begins with an account of the theory of abelian varieties needed for what follows. -
AN EXPLORATION of COMPLEX JACOBIAN VARIETIES Contents 1
AN EXPLORATION OF COMPLEX JACOBIAN VARIETIES MATTHEW WOOLF Abstract. In this paper, I will describe my thought process as I read in [1] about Abelian varieties in general, and the Jacobian variety associated to any compact Riemann surface in particular. I will also describe the way I currently think about the material, and any additional questions I have. I will not include material I personally knew before the beginning of the summer, which included the basics of algebraic and differential topology, real analysis, one complex variable, and some elementary material about complex algebraic curves. Contents 1. Abelian Sums (I) 1 2. Complex Manifolds 2 3. Hodge Theory and the Hodge Decomposition 3 4. Abelian Sums (II) 6 5. Jacobian Varieties 6 6. Line Bundles 8 7. Abelian Varieties 11 8. Intermediate Jacobians 15 9. Curves and their Jacobians 16 References 17 1. Abelian Sums (I) The first thing I read about were Abelian sums, which are sums of the form 3 X Z pi (1.1) (L) = !; i=1 p0 where ! is the meromorphic one-form dx=y, L is a line in P2 (the complex projective 2 3 2 plane), p0 is a fixed point of the cubic curve C = (y = x + ax + bx + c), and p1, p2, and p3 are the three intersections of the line L with C. The fact that C and L intersect in three points counting multiplicity is just Bezout's theorem. Since C is not simply connected, the integrals in the Abelian sum depend on the path, but there's no natural choice of path from p0 to pi, so this function depends on an arbitrary choice of path, and will not necessarily be holomorphic, or even Date: August 22, 2008. -
Lectures on the Combinatorial Structure of the Moduli Spaces of Riemann Surfaces
LECTURES ON THE COMBINATORIAL STRUCTURE OF THE MODULI SPACES OF RIEMANN SURFACES MOTOHICO MULASE Contents 1. Riemann Surfaces and Elliptic Functions 1 1.1. Basic Definitions 1 1.2. Elementary Examples 3 1.3. Weierstrass Elliptic Functions 10 1.4. Elliptic Functions and Elliptic Curves 13 1.5. Degeneration of the Weierstrass Elliptic Function 16 1.6. The Elliptic Modular Function 19 1.7. Compactification of the Moduli of Elliptic Curves 26 References 31 1. Riemann Surfaces and Elliptic Functions 1.1. Basic Definitions. Let us begin with defining Riemann surfaces and their moduli spaces. Definition 1.1 (Riemann surfaces). A Riemann surface is a paracompact Haus- S dorff topological space C with an open covering C = λ Uλ such that for each open set Uλ there is an open domain Vλ of the complex plane C and a homeomorphism (1.1) φλ : Vλ −→ Uλ −1 that satisfies that if Uλ ∩ Uµ 6= ∅, then the gluing map φµ ◦ φλ φ−1 (1.2) −1 φλ µ −1 Vλ ⊃ φλ (Uλ ∩ Uµ) −−−−→ Uλ ∩ Uµ −−−−→ φµ (Uλ ∩ Uµ) ⊂ Vµ is a biholomorphic function. Remark. (1) A topological space X is paracompact if for every open covering S S X = λ Uλ, there is a locally finite open cover X = i Vi such that Vi ⊂ Uλ for some λ. Locally finite means that for every x ∈ X, there are only finitely many Vi’s that contain x. X is said to be Hausdorff if for every pair of distinct points x, y of X, there are open neighborhoods Wx 3 x and Wy 3 y such that Wx ∩ Wy = ∅. -
Deligne Pairings and Families of Rank One Local Systems on Algebraic Curves
DELIGNE PAIRINGS AND FAMILIES OF RANK ONE LOCAL SYSTEMS ON ALGEBRAIC CURVES GERARD FREIXAS I MONTPLET AND RICHARD A. WENTWORTH Abstract. For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface. Contents 1. Introduction2 2. Relative Connections and Deligne Pairings5 2.1. Preliminary definitions5 2.2. Gauss-Manin invariant7 2.3. Deligne pairings, norm and trace9 2.4. Metrics and connections 10 3. Trace Connections and Intersection Connections 12 3.1. Weil reciprocity and trace connections 12 3.2. Reformulation in terms of Poincaré bundles 17 3.3. Intersection connections 20 4. Proof of the Main Theorems 23 4.1. The canonical extension: local description and properties 23 4.2. The canonical extension: uniqueness 29 4.3. Variant in the absence of rigidification 31 4.4. Relation between trace and intersection connections 32 arXiv:1507.02920v1 [math.DG] 10 Jul 2015 4.5. Curvatures 34 5. Examples and Applications 37 5.1. Reciprocity for trivial fibrations 37 5.2. Holomorphic extension of analytic torsion 40 5.3. -
The Riemann-Roch Theorem
THE RIEMANN-ROCH THEOREM GAL PORAT Abstract. These are notes for a talk which introduces the Riemann-Roch Theorem. We present the theorem in the language of line bundles and discuss its basic consequences, as well as an application to embeddings of curves in projective space. 1. Algebraic Curves and Riemann Surfaces A deep theorem due to Riemann says that every compact Riemann surface has a nonconstant meromorphic function. This leads to an equivalence {smooth projective complex algebraic curves} !{compact Riemann surfaces}. Topologically, all such Riemann surfaces are orientable compact manifolds, which are all genus g surfaces. Example 1.1. (1) The curve 1 corresponds to the Riemann sphere. PC (2) The (smooth projective closure of the) curve C : y2 = x3 − x corresponds to the torus C=(Z + iZ). Throughout the talk we will frequentely use both the point of view of algebraic curves and of Riemann surfaces in the context of the Riemann-Roch theorem. 2. Divisors Let C be a complex curve. A divisor of C is an element of the group Div(C) := ⊕P 2CZ. There is a natural map deg : Div(C) ! Z obtained by summing the coordinates. Functions f 2 K(C)× give rise to divisors on C. Indeed, there is a map div : K(C)× ! Div(C), given by setting X div(f) = vP (f)P: P 2C One can prove that deg(div(f)) = 0 for f 2 K(C)×; essentially this is a consequence of the argument principle, because the sum over all residues of any differential of a compact Riemann surface is equal to 0. -
Elliptic Factors in Jacobians of Hyperelliptic Curves with Certain Automorphism Groups Jennifer Paulhus
THE OPEN BOOK SERIES 1 ANTS X Proceedings of the Tenth Algorithmic Number Theory Symposium Elliptic factors in Jacobians of hyperelliptic curves with certain automorphism groups Jennifer Paulhus msp THE OPEN BOOK SERIES 1 (2013) Tenth Algorithmic Number Theory Symposium msp dx.doi.org/10.2140/obs.2013.1.487 Elliptic factors in Jacobians of hyperelliptic curves with certain automorphism groups Jennifer Paulhus We decompose the Jacobian varieties of hyperelliptic curves up to genus 20, defined over an algebraically closed field of characteristic zero, with reduced automorphism group A4, S4, or A5. Among these curves is a genus-4 curve 2 2 with Jacobian variety isogenous to E1 E2 and a genus-5 curve with Jacobian 5 variety isogenous to E , for E and Ei elliptic curves. These types of results have some interesting consequences for questions of ranks of elliptic curves and ranks of their twists. 1. Introduction Curves with Jacobian varieties that have many elliptic curve factors in their de- compositions up to isogeny have been studied in many different contexts. Ekedahl and Serre found examples of curves whose Jacobians split completely into elliptic curves (not necessarily isogenous to one another)[13] (see also[27],[14, §5]). In genus 2, Cardona showed connections between curves whose Jacobians have two isogenous elliptic curve factors and Q-curves of degree 2 and 3 [5]. There are applications of such curves to ranks of twists of elliptic curves[24], results on torsion[19], and cryptography[12]. Let JX denote the Jacobian variety of a curve X and let represent an isogeny between abelian varieties. -
Notes on Riemann Surfaces
NOTES ON RIEMANN SURFACES DRAGAN MILICIˇ C´ 1. Riemann surfaces 1.1. Riemann surfaces as complex manifolds. Let M be a topological space. A chart on M is a triple c = (U, ϕ) consisting of an open subset U M and a homeomorphism ϕ of U onto an open set in the complex plane C. The⊂ open set U is called the domain of the chart c. The charts c = (U, ϕ) and c′ = (U ′, ϕ′) on M are compatible if either U U ′ = or U U ′ = and ϕ′ ϕ−1 : ϕ(U U ′) ϕ′(U U ′) is a bijective holomorphic∩ ∅ function∩ (hence ∅ the inverse◦ map is∩ also holomorphic).−→ ∩ A family of charts on M is an atlas of M if the domains of charts form a covering of MA and any two charts in are compatible. Atlases and of M are compatibleA if their union is an atlas on M. This is obviously anA equivalenceB relation on the set of all atlases on M. Each equivalence class of atlases contains the largest element which is equal to the union of all atlases in this class. Such atlas is called saturated. An one-dimensional complex manifold M is a hausdorff topological space with a saturated atlas. If M is connected we also call it a Riemann surface. Let M be an Riemann surface. A chart c = (U, ϕ) is a chart around z M if z U. We say that it is centered at z if ϕ(z) = 0. ∈ ∈Let M and N be two Riemann surfaces. A continuous map F : M N is a holomorphic map if for any two pairs of charts c = (U, ϕ) on M and d =−→ (V, ψ) on N such that F (U) V , the mapping ⊂ ψ F ϕ−1 : ϕ(U) ϕ(V ) ◦ ◦ −→ is a holomorphic function. -
6. the Riemann Sphere It Is Sometimes Convenient to Add a Point at Infinity ∞ to the Usual Complex Plane to Get the Extended Complex Plane
6. The Riemann sphere It is sometimes convenient to add a point at infinity 1 to the usual complex plane to get the extended complex plane. Definition 6.1. The extended complex plane, denoted P1, is simply the union of C and the point at infinity. It is somewhat curious that when we add points at infinity to the reals we add two points ±∞ but only only one point for the complex numbers. It is rare in geometry that things get easier as you increase the dimension. One very good way to understand the extended complex plane is to realise that P1 is naturally in bijection with the unit sphere: Definition 6.2. The Riemann sphere is the unit sphere in R3: 2 3 2 2 2 S = f (x; y; z) 2 R j x + y + z = 1 g: To make a correspondence with the sphere and the plane is simply to make a map (a real map not a function). We define a function 2 3 F : S − f(0; 0; 1)g −! C ⊂ R as follows. Pick a point q = (x; y; z) 2 S2, a point of the unit sphere, other than the north pole p = N = (0; 0; 1). Connect the point p to the point q by a line. This line will meet the plane z = 0, 2 f (x; y; 0) j (x; y) 2 R g in a unique point r. We then identify r with a point F (q) = x + iy 2 C in the usual way. F is called stereographic projection. -
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.