Algebraic Methods for Dynamical Systems and Optimisation
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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. -
[Math.AG] 1 Nov 2005 Where Uoopimgop Hscntuto Ensahlmrhcperio Holomorphic a Defines Construction This Group
A COMPACTIFICATION OF M3 VIA K3 SURFACES MICHELA ARTEBANI Abstract. S. Kond¯odefined a birational period map from the moduli space of genus three curves to a moduli space of degree four polarized K3 surfaces. In this paper we extend the period map to a surjective morphism on a suitable compactification of M3 and describe its geometry. Introduction 14 Let V =| OP2 (4) |=∼ P be the space of plane quartics and V0 be the open subvariety of smooth curves. The degree four cyclic cover of the plane branched along a curve C ∈ V0 is a K3 surface equipped with an order four non-symplectic automorphism group. This construction defines a holomorphic period map: P0 : V0 −→ M, where V0 is the geometric quotient of V0 by the action of P GL(3) and M is a moduli space of polarized K3 surfaces. In [15] S. Kond¯oshows that P0 gives an isomorphism between V0 and the com- plement of two irreducible divisors Dn, Dh in M. Moreover, he proves that the generic points in Dn and Dh correspond to plane quartics with a node and to smooth hyperelliptic genus three curves respectively. The moduli space M is an arithmetic quotient of a six dimensional complex ball, hence a natural compactification is given by the Baily-Borel compactification M∗ (see [1]). On the other hand, geometric invariant theory provides a compact pro- jective variety V containing V0 as a dense subset, given by the categorical quotient of the semistable locus in V for the natural action of P GL(3). In this paper we prove that the map P0 can be extended to a holomorphic surjective map P : V −→M∗ on the blowing-up V of V in the point v0 corresponding to the orbit of double arXiv:math/0511031v1 [math.AG] 1 Nov 2005 e conics. -
Real Rank Two Geometry Arxiv:1609.09245V3 [Math.AG] 5
Real Rank Two Geometry Anna Seigal and Bernd Sturmfels Abstract The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two. 1 Introduction Low-rank approximation of tensors is a fundamental problem in applied mathematics [3, 6]. We here approach this problem from the perspective of real algebraic geometry. Our goal is to give an exact semi-algebraic description of the set of tensors of real rank two and to characterize its boundary. This complements the results on tensors of non-negative rank two presented in [1], and it offers a generalization to the setting of arbitrary varieties, following [2]. A familiar example is that of 2 × 2 × 2-tensors (xijk) with real entries. Such a tensor lies in the closure of the real rank two tensors if and only if the hyperdeterminant is non-negative: 2 2 2 2 2 2 2 2 x000x111 + x001x110 + x010x101 + x011x100 + 4x000x011x101x110 + 4x001x010x100x111 −2x000x001x110x111 − 2x000x010x101x111 − 2x000x011x100x111 (1) −2x001x010x101x110 − 2x001x011x100x110 − 2x010x011x100x101 ≥ 0: If this inequality does not hold then the tensor has rank two over C but rank three over R. To understand this example geometrically, consider the Segre variety X = Seg(P1 × P1 × P1), i.e. the set of rank one tensors, regarded as points in the projective space P7 = 2 2 2 7 arXiv:1609.09245v3 [math.AG] 5 Apr 2017 P(C ⊗ C ⊗ C ). -
Shadow of a Cubic Surface
Faculteit B`etawetenschappen Shadow of a cubic surface Bachelor Thesis Rein Janssen Groesbeek Wiskunde en Natuurkunde Supervisors: Dr. Martijn Kool Departement Wiskunde Dr. Thomas Grimm ITF June 2020 Abstract 3 For a smooth cubic surface S in P we can cast a shadow from a point P 2 S that does not lie on one of the 27 lines of S onto a hyperplane H. The closure of this shadow is a smooth quartic curve. Conversely, from every smooth quartic curve we can reconstruct a smooth cubic surface whose closure of the shadow is this quartic curve. We will also present an algorithm to reconstruct the cubic surface from the bitangents of a quartic curve. The 27 lines of S together with the tangent space TP S at P are in correspondence with the 28 bitangents or hyperflexes of the smooth quartic shadow curve. Then a short discussion on F-theory is given to relate this geometry to physics. Acknowledgements I would like to thank Martijn Kool for suggesting the topic of the shadow of a cubic surface to me and for the discussions on this topic. Also I would like to thank Thomas Grimm for the suggestions on the applications in physics of these cubic surfaces. Finally I would like to thank the developers of Singular, Sagemath and PovRay for making their software available for free. i Contents 1 Introduction 1 2 The shadow of a smooth cubic surface 1 2.1 Projection of the first polar . .1 2.2 Reconstructing a cubic from the shadow . .5 3 The 27 lines and the 28 bitangents 9 3.1 Theorem of the apparent boundary . -
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. -
Diophantine Problems in Polynomial Theory
Diophantine Problems in Polynomial Theory by Paul David Lee B.Sc. Mathematics, The University of British Columbia, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The College of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) August 2011 c Paul David Lee 2011 Abstract Algebraic curves and surfaces are playing an increasing role in mod- ern mathematics. From the well known applications to cryptography, to computer vision and manufacturing, studying these curves is a prevalent problem that is appearing more often. With the advancement of computers, dramatic progress has been made in all branches of algebraic computation. In particular, computer algebra software has made it much easier to find rational or integral points on algebraic curves. Computers have also made it easier to obtain rational parametrizations of certain curves and surfaces. Each algebraic curve has an associated genus, essentially a classification, that determines its topological structure. Advancements on methods and theory on curves of genus 0, 1 and 2 have been made in recent years. Curves of genus 0 are the only algebraic curves that you can obtain a rational parametrization for. Curves of genus 1 (also known as elliptic curves) have the property that their rational points have a group structure and thus one can call upon the massive field of group theory to help with their study. Curves of higher genus (such as genus 2) do not have the background and theory that genus 0 and 1 do but recent advancements in theory have rapidly expanded advancements on the topic. -
Tropical Curves
Tropical Curves Nathan Pflueger 24 February 2011 Abstract A tropical curve is a graph with specified edge lengths, some of which may be infinite. Various facts and attributes about algebraic curves have analogs for tropical curves. In this article, we focus on divisors and linear series, and prove the Riemann-Roch formula for divisors on tropical curves. We describe two ways in which algebraic curves may be transformed into tropical curves: by aboemas and by specialization on arithmetic surfaces. We discuss how the study of linear series on tropical curves can be used to obtain results about linear series on algebraic curves, and summarize several recent applications. Contents 1 Introduction 2 2 From curves to graphs 3 2.1 Amoebas of plane curves . .3 2.2 Curves over the field of Puiseux series . .4 2.3 Specialization . .5 3 Metric graphs and tropical curves 6 4 Divisors and linear equivalence on tropical curves 9 4.1 The Riemann-Roch criterion . 12 4.2 Tropical Riemann-Roch . 14 5 Tropical plane curves 17 5.1 Tropical algebra and tropical projective space . 17 5.2 Tropical curves in R2 ................................... 18 5.3 Calculation of the genus . 23 5.4 Stable intersection and the tropical B´ezouttheorem . 24 5.5 Classical B´ezoutfrom tropical B´ezout . 29 5.6 Enumerative geometry of tropical plane curves . 31 6 Tropical curves via specialization 32 6.1 The specialization map and specialization lemma . 32 6.2 The canonical divisor of a graph is canonical . 34 6.3 A tropical proof of the Brill-Noether theorem . 34 1 1 Introduction The origins of tropical geometry lie in the study of tropical algebra, whose basic object is the set R [ {−∞} equipped with the operations x ⊕ y = max(x; y) and x ⊗ y = x + y. -
Geometry of Algebraic Curves
Geometry of Algebraic Curves Fall 2011 Course taught by Joe Harris Notes by Atanas Atanasov One Oxford Street, Cambridge, MA 02138 E-mail address: [email protected] Contents Lecture 1. September 2, 2011 6 Lecture 2. September 7, 2011 10 2.1. Riemann surfaces associated to a polynomial 10 2.2. The degree of KX and Riemann-Hurwitz 13 2.3. Maps into projective space 15 2.4. An amusing fact 16 Lecture 3. September 9, 2011 17 3.1. Embedding Riemann surfaces in projective space 17 3.2. Geometric Riemann-Roch 17 3.3. Adjunction 18 Lecture 4. September 12, 2011 21 4.1. A change of viewpoint 21 4.2. The Brill-Noether problem 21 Lecture 5. September 16, 2011 25 5.1. Remark on a homework problem 25 5.2. Abel's Theorem 25 5.3. Examples and applications 27 Lecture 6. September 21, 2011 30 6.1. The canonical divisor on a smooth plane curve 30 6.2. More general divisors on smooth plane curves 31 6.3. The canonical divisor on a nodal plane curve 32 6.4. More general divisors on nodal plane curves 33 Lecture 7. September 23, 2011 35 7.1. More on divisors 35 7.2. Riemann-Roch, finally 36 7.3. Fun applications 37 7.4. Sheaf cohomology 37 Lecture 8. September 28, 2011 40 8.1. Examples of low genus 40 8.2. Hyperelliptic curves 40 8.3. Low genus examples 42 Lecture 9. September 30, 2011 44 9.1. Automorphisms of genus 0 an 1 curves 44 9.2. -
Foundations of Algebraic Geometry Class 41
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 41 RAVI VAKIL CONTENTS 1. Normalization 1 2. Extending maps to projective schemes over smooth codimension one points: the “clear denominators” theorem 5 Welcome back! Let's now use what we have developed to study something explicit — curves. Our motivating question is a loose one: what are the curves, by which I mean nonsingular irreducible separated curves, finite type over a field k? In other words, we'll be dealing with geometry, although possibly over a non-algebraically closed field. Here is an explicit question: are all curves (say reduced, even non-singular, finite type over given k) isomorphic? Obviously not: some are affine, and some (such as P1) are not. So to simplify things — and we'll further motivate this simplification in Class 42 — are all projective curves isomorphic? Perhaps all nonsingular projective curves are isomorphic to P1? Once again the answer is no, but the proof is a bit subtle: we've defined an invariant, the genus, and shown that P1 has genus 0, and that there exist nonsingular projective curves of non-zero genus. Are all (nonsingular) genus 0 curves isomorphic to P1? We know there exist nonsingular genus 1 curves (plane cubics) — is there only one? If not, “how many” are there? In order to discuss interesting questions like these, we'll develop some theory. We first show a useful result that will help us focus our attention on the projective case. 1. NORMALIZATION I now want to tell you how to normalize a reduced Noetherian scheme, which is roughly how best to turn a scheme into a normal scheme. -
Triple Conformal Geometric Algebra for Cubic Plane Curves
Received 13 February 2017; Revised 3 July 2018 (revised preprint with corrections) ; Accepted 22 August 2017 DOI: 10.1002/mma.4597 is revision 18 Sep 2017, published in Mathematical Methods in the Applied Sciences, 41(11)4088–4105, 30 July 2018, Special Issue: ENGAGE SPECIAL ISSUE PAPER Triple Conformal Geometric Algebra for Cubic Plane Curves Robert Benjamin Easter1 | Eckhard Hitzer2 1Bangkok, Thailand. Email: [email protected] Summary 2College of Liberal Arts, International The Triple Conformal Geometric Algebra (TCGA) for the Euclidean R2-plane ex- Christian University, 3-10-2 Osawa, 181-8585 Mitaka, Tokyo, Japan. Email: tends CGA as the product of three orthogonal CGAs, and thereby the representation [email protected] of geometric entities to general cubic plane curves and certain cyclidic (or roulette) Communicated by: Dietmar Hildenbrand MSC Primary: 15A66; quartic, quintic, and sextic plane curves. The plane curve entities are 3-vectors that MSC Secondary: 14H50; 53A30; linearize the representation of non-linear curves, and the entities are inner product Correspondence null spaces (IPNS) with respect to all points on the represented curves. Each IPNS Eckhard Hitzer, College of Liberal Arts, entity also has a dual geometric outer product null space (OPNS) form. Orthogonal or International Christian University, 3-10-2 conformal (angle-preserving) operations (as versors) are valid on all TCGA entities Osawa, 181-8585 Mitaka, Tokyo, Japan. Email: [email protected] for inversions in circles, reflections in lines, and, by compositions thereof, isotropic dilations from a given center point, translations, and rotations around arbitrary points in the plane. A further dimensional extension of TCGA, also provides a method for anisotropic dilations. -
Limits of Pluri–Tangent Planes to Quartic Surfaces Ciro Ciliberto and Thomas Dedieu
Limits of pluri–tangent planes to quartic surfaces Ciro Ciliberto and Thomas Dedieu Abstract. We describe, for various degenerations S → ∆ of quartic K3 surfaces over the complex unit disk (e.g., to the union of four general planes, and to a general Kummer surface), the limits ∗ as t ∈ ∆ tends to 0 of the Severi varieties Vδ(St), parametrizing irreducible δ-nodal plane sections of St. We give applications of this to (i) the counting of plane nodal curves through base points in special position, (ii) the irreducibility of Severi varieties of a general quartic surface, and (iii) the monodromy of the universal family of rational curves on quartic K3 surfaces. Contents 1 Conventions 4 2 Limit linear systems and limit Severi varieties 4 3 Auxiliary results 10 4 Degeneration to a tetrahedron 11 5 Other degenerations 28 6 Kummer quartic surfaces in P3 30 7 Degeneration to a Kummer surface 33 8 Plane quartics curves through points in special position 41 9 Application to the irreducibility of Severi varieties and to the monodromy action 46 Introduction Our objective in this paper is to study the following: Question A Let f : S → ∆ be a projective family of surfaces of degree d in P3, with S a smooth −1 threefold, and ∆ the complex unit disc (usually called a degeneration of the general St := f (t), for t 6= 0, which is a smooth surface, to the central fibre S0, which is in general supposed to be singular). What are the limits of tangent, bitangent, and tritangent planes to St, for t 6=0, as t tends to 0? Similar questions make sense also for degenerations of plane curves, and we refer to [25, pp. -
Algebraic Plane Curves Deposited by the Faculty of Graduate Studies and Research
PLUCKER'3 : NUMBERS: ALGEBRAIC PLANE CURVES DEPOSITED BY THE FACULTY OF GRADUATE STUDIES AND RESEARCH Iw • \TS • \t£S ACC. NO UNACC.'»"1928 PLITCOR'S FOOTERS IE THE THEORY 0? ALGEBRAIC PLAITS CURVES A Thesis presented in partial fulfilment for the degree of -Master of Arts, Department of Mathematics. April 28, 1928. by ALICiD WI1LARD TURIIPP. Acknowledgment Any commendation which this thesis may merit, is in large measure due to Dr. C..T. Sullivan, Peter Re&path Professor of Pure Mathematics; and I take this opportunity to express my indebtedness to him. April 25, 1928. I IT D A :: IITTPODIICTIOI? SECTIOH I. • pp.l-S4 Algebraic Curve defined. Intersection of two curves. Singular points on curves. Conditions for multiple points. Conditions determining an n-ic. Maximum number of double points. Examples. SECTION II ...... pp.25-39 Class of curve. Tangerfcial equations. Polar reciprocation. Singularities on a curve and its reciprocal. Superlinear branches, with application to intersections of two curves at singular points. Examples. 3SCTI0IT III pp.40-52 Polar Curves. Intersections of Curve and Pirst Polar. Hessian.Intersections of Hessian and Curve. Examples. 3ECTI0TT IV . .................. pp.52-69 Pliicker's numbers and Equations for simple singularities. Deficiency. Unicursal Curves. Classification of cubies and quartics. Extension of Plucker's relations to k-ple points with distinct tangents, and to ordinary superlinear branch points. Short discussion of Higher Singularities. Examples. I PhTJCPEirS PTE.3EE3 IE PEE EHSOHY CF ALGEBRAIC Ph-TITE CUPYE3. IPPEODUCPIOH Pith the advent of the nineteenth century,a new era dawned in the progress of analytic geometry. The appearance of poiiceletTs, "Traite des proprietes project ives des figures", in 1822, really initiated modern geometry.