Group Actions, Divisors, and Plane 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. -
Arxiv:1912.10980V2 [Math.AG] 28 Jan 2021 6
Automorphisms of real del Pezzo surfaces and the real plane Cremona group Egor Yasinsky* Universität Basel Departement Mathematik und Informatik Spiegelgasse 1, 4051 Basel, Switzerland ABSTRACT. We study automorphism groups of real del Pezzo surfaces, concentrating on finite groups acting with invariant Picard number equal to one. As a result, we obtain a vast part of classification of finite subgroups in the real plane Cremona group. CONTENTS 1. Introduction 2 1.1. The classification problem2 1.2. G-surfaces3 1.3. Some comments on the conic bundle case4 1.4. Notation and conventions6 2. Some auxiliary results7 2.1. A quick look at (real) del Pezzo surfaces7 2.2. Sarkisov links8 2.3. Topological bounds9 2.4. Classical linear groups 10 3. Del Pezzo surfaces of degree 8 10 4. Del Pezzo surfaces of degree 6 13 5. Del Pezzo surfaces of degree 5 16 arXiv:1912.10980v2 [math.AG] 28 Jan 2021 6. Del Pezzo surfaces of degree 4 18 6.1. Topology and equations 18 6.2. Automorphisms 20 6.3. Groups acting minimally on real del Pezzo quartics 21 7. Del Pezzo surfaces of degree 3: cubic surfaces 28 Sylvester non-degenerate cubic surfaces 34 7.1. Clebsch diagonal cubic 35 *[email protected] Keywords: Cremona group, conic bundle, del Pezzo surface, automorphism group, real algebraic surface. 1 2 7.2. Cubic surfaces with automorphism group S4 36 Sylvester degenerate cubic surfaces 37 7.3. Equianharmonic case: Fermat cubic 37 7.4. Non-equianharmonic case 39 7.5. Non-cyclic Sylvester degenerate surfaces 39 8. Del Pezzo surfaces of degree 2 40 9. -
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. -
Combination of Cubic and Quartic Plane Curve
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 43-53 www.iosrjournals.org Combination of Cubic and Quartic Plane Curve C.Dayanithi Research Scholar, Cmj University, Megalaya Abstract The set of complex eigenvalues of unistochastic matrices of order three forms a deltoid. A cross-section of the set of unistochastic matrices of order three forms a deltoid. The set of possible traces of unitary matrices belonging to the group SU(3) forms a deltoid. The intersection of two deltoids parametrizes a family of Complex Hadamard matrices of order six. The set of all Simson lines of given triangle, form an envelope in the shape of a deltoid. This is known as the Steiner deltoid or Steiner's hypocycloid after Jakob Steiner who described the shape and symmetry of the curve in 1856. The envelope of the area bisectors of a triangle is a deltoid (in the broader sense defined above) with vertices at the midpoints of the medians. The sides of the deltoid are arcs of hyperbolas that are asymptotic to the triangle's sides. I. Introduction Various combinations of coefficients in the above equation give rise to various important families of curves as listed below. 1. Bicorn curve 2. Klein quartic 3. Bullet-nose curve 4. Lemniscate of Bernoulli 5. Cartesian oval 6. Lemniscate of Gerono 7. Cassini oval 8. Lüroth quartic 9. Deltoid curve 10. Spiric section 11. Hippopede 12. Toric section 13. Kampyle of Eudoxus 14. Trott curve II. Bicorn curve In geometry, the bicorn, also known as a cocked hat curve due to its resemblance to a bicorne, is a rational quartic curve defined by the equation It has two cusps and is symmetric about the y-axis. -
Computer Graphics Lines and Planes Motivations Tekla T´Oth Lines [email protected] Planes
Introduction Computer Graphics Lines and planes Motivations Tekla T´oth Lines [email protected] Planes E¨otv¨osLor´andTudom´anyegyetem Faculty of Informatics Simple shapes 2019 fall (2019-2020-1) Curves Surfaces Motivation Curves and surfaces I Informally, the curves and surfaces are 'special' subsets of I We can now represent the points of the plane or space by space - i.e. they are sets of points numbers (their coordinates) I How can we define these - usually infinite - sets? I How can we represent 'nice' sets of points, e.g. a line in the I explicit: y = f (x) ! what happens when the curve should plane or a plane in space? 'head back'? x(t) parametric: p(t) = ; t 2 I We seek the answer to this in the Cartesian coordinate system I y(t) R I implicit: x 2 + y 2 − 9 = 0 The 'old school' explicit line Line with a point and normal T I Let p(px ; py ) be a point on the line and n = [nx ; ny ] 6= 0 a vector, a normal perpendicular to the line. I Highschool: y = mx + b I Then all x(x; y) points of the plane that satisfy the following are exactly the points of the line: I Problem: vertical lines! hx − p; ni = 0 (x − px )nx + (y − py )ny = 0 I Two half-planes: hx0 − p; ni < 0 ´es hx0 − p; ni > 0 The homogeneous implicit equation of the line on the plane Homogeneous implicit equation with determinant I Let p(px ; py ) and q(qx ; qy ) be two distinct points on the line. -
Effective Methods for Plane Quartics, Their Theta Characteristics and The
Effective methods for plane quartics, their theta characteristics and the Scorza map Giorgio Ottaviani Abstract Notes for the workshop on \Plane quartics, Scorza map and related topics", held in Catania, January 19-21, 2016. The last section con- tains eight Macaulay2 scripts on theta characteristics and the Scorza map, with a tutorial. The first sections give an introduction to these scripts. Contents 1 Introduction 2 1.1 How to write down plane quartics and their theta characteristics 2 1.2 Clebsch and L¨urothquartics . 4 1.3 The Scorza map . 5 1.4 Description of the content . 5 1.5 Summary of the eight M2 scripts presented in x9 . 6 1.6 Acknowledgements . 7 2 Apolarity, Waring decompositions 7 3 The Aronhold invariant of plane cubics 8 4 Three descriptions of an even theta characteristic 11 4.1 The symmetric determinantal description . 11 4.2 The sextic model . 11 4.3 The (3; 3)-correspondence . 13 5 The Aronhold covariant of plane quartics and the Scorza map. 14 1 6 Contact cubics and contact triangles 15 7 The invariant ring of plane quartics 17 8 The link with the seven eigentensors of a plane cubic 18 9 Eight algorithms and Macaulay2 scripts, with a tutorial 19 1 Introduction 1.1 How to write down plane quartics and their theta characteristics Plane quartics make a relevant family of algebraic curves because their plane embedding is the canonical embedding. As a byproduct, intrinsic and pro- jective geometry are strictly connected. It is not a surprise that the theta characteristics of a plane quartic, being the 64 square roots of the canonical bundle, show up in many projective constructions. -
The Group of Automorphisms of the Heisenberg Curve
THE GROUP OF AUTOMORPHISMS OF THE HEISENBERG CURVE JANNIS A. ANTONIADIS AND ARISTIDES KONTOGEORGIS ABSTRACT. The Heisenberg curve is defined to be the curve corresponding to an exten- sion of the projective line by the Heisenberg group modulo n, ramified above three points. This curve is related to the Fermat curve and its group of automorphisms is studied. Also we give an explicit equation for the curve C3. 1. INTRODUCTION Probably the most famous curve in number theory is the Fermat curve given by affine equation n n Fn : x + y =1. This curve can be seen as ramified Galois cover of the projective line with Galois group a b Z/nZ Z/nZ with action given by σa,b : (x, y) (ζ x, ζ y) where (a,b) Z/nZ Z/nZ.× The ramified cover 7→ ∈ × π : F P1 n → has three ramified points and the cover F 0 := F π−1( 0, 1, ) π P1 0, 1, , n n − { ∞} −→ −{ ∞} is a Galois topological cover. We can see the hyperbolic space H as the universal covering space of P1 0, 1, . The Galois group of the above cover is isomorphic to the free −{ ∞} group F2 in two generators, and a suitable realization of this group in our setting is the group ∆ which is the subgroup of SL(2, Z) PSL(2, R) generated by the elements a = 1 2 1 0 ⊆ , b = and π(P1 0, 1, , x ) = ∆. Related to the group ∆ is the 0 1 2 1 −{ ∞} 0 ∼ modular group a b Γ(2) = γ = SL(2, Z): γ 1 mod2 , c d ∈ ≡ 2 H P1 which is isomorphic to I ∆ while Γ(2) ∼= 0, 1, . -
Arxiv:1012.2020V1 [Math.CV]
TRANSITIVITY ON WEIERSTRASS POINTS ZOË LAING AND DAVID SINGERMAN 1. Introduction An automorphism of a Riemann surface will preserve its set of Weier- strass points. In this paper, we search for Riemann surfaces whose automorphism groups act transitively on the Weierstrass points. One well-known example is Klein’s quartic, which is known to have 24 Weierstrass points permuted transitively by it’s automorphism group, PSL(2, 7) of order 168. An investigation of when Hurwitz groups act transitively has been made by Magaard and Völklein [19]. After a section on the preliminaries, we examine the transitivity property on several classes of surfaces. The easiest case is when the surface is hy- perelliptic, and we find all hyperelliptic surfaces with the transitivity property (there are infinitely many of them). We then consider surfaces with automorphism group PSL(2, q), Weierstrass points of weight 1, and other classes of Riemann surfaces, ending with Fermat curves. Basically, we find that the transitivity property property seems quite rare and that the surfaces we have found with this property are inter- esting for other reasons too. 2. Preliminaries Weierstrass Gap Theorem ([6]). Let X be a compact Riemann sur- face of genus g. Then for each point p ∈ X there are precisely g integers 1 = γ1 < γ2 <...<γg < 2g such that there is no meromor- arXiv:1012.2020v1 [math.CV] 9 Dec 2010 phic function on X whose only pole is one of order γj at p and which is analytic elsewhere. The integers γ1,...,γg are called the gaps at p. The complement of the gaps at p in the natural numbers are called the non-gaps at p. -
Moduli of Curves 1
THE MODULI SPACE OF CURVES In this section, we will give a sketch of the construction of the moduli space Mg of curves of genus g and the closely related moduli space Mg;n of n-pointed curves of genus g using two different approaches. Throughout this section we always assume that 2g − 2 + n > 0. In the first approach, we will embed curves in a projective space PN using a sufficiently high power n of their dualizing sheaf. Then a locus in the Hilbert scheme parameterizes n-canonically embedded curves of genus g. The automorphism group PGL(N + 1) acts on the Hilbert scheme. Using Mumford's Geometric Invariant Theory, one can take the quotient of the appropriate locus in the Hilbert scheme by the action of PGL(N + 1) to construct Mg. In the second approach, one first constructs the moduli space of curves as a Deligne-Mumford stack. One then exhibits an ample line bundle on the coarse moduli scheme of this stack. 1. Basics about curves We begin by collecting basic facts and definitions about stable curves. A curve singularity (C; p) is called a node if locally analytically the singularity is isomophic to the plane curve singularity xy = 0. A curve C is called at-worst-nodal or more simply nodal if the only singularities of C are nodes. The dualizing sheaf !C of an at-worst-nodal curve C is an invertible sheaf that has a simple description. Let ν : Cν ! C be the nor- malization of the curve C. Let p1; : : : ; pδ be the nodes of C and let −1 fri; sig = ν (pi). -
Two Cases of Fermat's Last Theorem Using Descent on Elliptic Curves
A. M. Viergever Two cases of Fermat's Last Theorem using descent on elliptic curves Bachelor thesis 22 June 2018 Thesis supervisor: dr. R.M. van Luijk Leiden University Mathematical Institute Contents Introduction2 1 Some important theorems and definitions4 1.1 Often used notation, definitions and theorems.................4 1.2 Descent by 2-isogeny...............................5 1.3 Descent by 3-isogeny...............................8 2 The case n = 4 10 2.1 The standard proof................................ 10 2.2 Descent by 2-isogeny............................... 11 2.2.1 C and C0 are elliptic curves....................... 11 2.2.2 The descent-method applied to E .................... 13 2.2.3 Relating the descent method and the classical proof.......... 14 2.2.4 A note on the automorphisms of C ................... 16 2.3 The method of descent by 2-isogeny for a special class of elliptic curves... 17 2.3.1 Addition formulas............................. 17 2.3.2 The method of 2-descent on (Ct; O)................... 19 2.4 Twists and φ-coverings.............................. 22 2.4.1 Twists................................... 22 2.4.2 A φ-covering................................ 25 3 The case n = 3 26 3.1 The standard proof................................ 26 3.2 Descent by 3-isogeny............................... 28 3.2.1 The image of q .............................. 28 3.2.2 The descent method applied to E .................... 30 3.2.3 Relating the descent method and the classical proof.......... 32 3.2.4 A note on the automorphisms of C ................... 33 3.3 Twists and φ-coverings.............................. 34 3.3.1 The case d 2 (Q∗)2 ............................ 34 3.3.2 The case d2 = (Q∗)2 ........................... -
Singularities of Integrable Systems and Nodal Curves
Singularities of integrable systems and nodal curves Anton Izosimov∗ Abstract The relation between integrable systems and algebraic geometry is known since the XIXth century. The modern approach is to represent an integrable system as a Lax equation with spectral parameter. In this approach, the integrals of the system turn out to be the coefficients of the characteristic polynomial χ of the Lax matrix, and the solutions are expressed in terms of theta functions related to the curve χ = 0. The aim of the present paper is to show that the possibility to write an integrable system in the Lax form, as well as the algebro-geometric technique related to this possibility, may also be applied to study qualitative features of the system, in particular its singularities. Introduction It is well known that the majority of finite dimensional integrable systems can be written in the form d L(λ) = [L(λ),A(λ)] (1) dt where L and A are matrices depending on the time t and additional parameter λ. The parameter λ is called a spectral parameter, and equation (1) is called a Lax equation with spectral parameter1. The possibility to write a system in the Lax form allows us to solve it explicitly by means of algebro-geometric technique. The algebro-geometric scheme of solving Lax equations can be briefly described as follows. Let us assume that the dependence on λ is polynomial. Then, with each matrix polynomial L, there is an associated algebraic curve C(L)= {(λ, µ) ∈ C2 | det(L(λ) − µE) = 0} (2) called the spectral curve. -
Log Minimal Model Program for the Moduli Space of Stable Curves: the Second Flip
LOG MINIMAL MODEL PROGRAM FOR THE MODULI SPACE OF STABLE CURVES: THE SECOND FLIP JAROD ALPER, MAKSYM FEDORCHUK, DAVID ISHII SMYTH, AND FREDERICK VAN DER WYCK Abstract. We prove an existence theorem for good moduli spaces, and use it to construct the second flip in the log minimal model program for M g. In fact, our methods give a uniform self-contained construction of the first three steps of the log minimal model program for M g and M g;n. Contents 1. Introduction 2 Outline of the paper 4 Acknowledgments 5 2. α-stability 6 2.1. Definition of α-stability6 2.2. Deformation openness 10 2.3. Properties of α-stability 16 2.4. αc-closed curves 18 2.5. Combinatorial type of an αc-closed curve 22 3. Local description of the flips 27 3.1. Local quotient presentations 27 3.2. Preliminary facts about local VGIT 30 3.3. Deformation theory of αc-closed curves 32 3.4. Local VGIT chambers for an αc-closed curve 40 4. Existence of good moduli spaces 45 4.1. General existence results 45 4.2. Application to Mg;n(α) 54 5. Projectivity of the good moduli spaces 64 5.1. Main positivity result 67 5.2. Degenerations and simultaneous normalization 69 5.3. Preliminary positivity results 74 5.4. Proof of Theorem 5.5(a) 79 5.5. Proof of Theorem 5.5(b) 80 Appendix A. 89 References 92 1 2 ALPER, FEDORCHUK, SMYTH, AND VAN DER WYCK 1. Introduction In an effort to understand the canonical model of M g, Hassett and Keel introduced the log minimal model program (LMMP) for M .