DOCSLIB.ORG

Algebraic Geometry I

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebraic Geometry I GEIR ELLINGSRUD NOTES FOR MA4210— ALGEBRAIC GEOMETRY I Contents 1 Algebraic sets and the Nullstellensatz 7 Fields and the affine space 8 Closed algebraic sets 8 The Nullstellensatz 11 Hilbert’s Nullstellensatz—proofs 13 Figures and intuition 15 A second proof of the Nullstellensatz 16 2 Zariski topolgies 21 The Zariski topology 21 Irreducible topological spaces 23 Polynomial maps between algebraic sets 31 3 Sheaves and varities 37 Sheaves of rings 37 Functions on irreducible algebraic sets 40 The definition of a variety 43 Morphisms between prevarieties 46 The Hausdorff axiom 48 Products of varieties 50 4 Projective varieties 59 The projective spaces Pn 60 The projective Nullstellensatz 68 2 Global regular functions on projective varieties 70 Morphisms from quasi projective varieties 72 Two important classes of subvarieties 76 The Veronese embeddings 77 The Segre embeddings 79 5 Dimension 83 Definition of the dimension 84 Finite polynomial maps 88 Noether’s Normalization Lemma 93 Krull’s Principal Ideal Theorem 97 Applications to intersections 102 Appendix: Proof of the Geometric Principal Ideal Theorem 105 6 Rational Maps and Curves 111 Rational and birational maps 112 Curves 117 7 Structure of maps 127 Generic structure of morhisms 127 Properness of projectives 131 Finite maps 134 Curves over regular curves 140 8 Bézout’s theorem 143 Bézout’s Theorem 144 The local multiplicity 145 Proof of Bezout’s theorem 148 8.3.1 A general lemma 150 Appendix: Depth, regular sequences and unmixedness 152 Appendix: Some graded algebra 158 3 9 Non-singular varieties 165 Regular local rings 165 The Jacobian criterion 166 9.2.1 The projective cse 167 CONTENTS 5 These notes ere just informal extensions of the lectures I gave last year. As the course developeds I’ll now and then posted new notes on the course’s website, but this will certainly happen with irregular intervals. The idea with the notes was to give additional comments and examples which hopefully made reading of the book and the digestion of the lectures easier; and hopefully widened the students mathematical horizon. It seems that other lectures are interested in the notes, so I try to upgrade them—correct misprints and not to the least give correct proofs of all theorems (important buisiness!!!). This is an ongoing process and the present version is still preliminary. As the students last year survived the notes in the then shaky condition, I am confident that students this year will survive as well; and still better (or more humbly less bad) version are coming! GE Lecture 1 Algebraic sets and the Nullstellensatz Hot themes in Lecture 1: The correspondence between ideals and algebraic sets—weak and strong versions of Hilbert’s Nullstellensatz—the Rabinowitsch trick—two proofs of the Nullstellensatz, one elementary, and another totally different—radical ideals—intuition, drawings and figures. Algebraic geometry has many ramifications, but roughly speaking there are two main branches. One could be called the “geometric” branch where the geometry is the main objective. One studies geometric objects like curves, surfaces, threefolds and varieties of higher dimensions, defined by polyno- mials (or more generally algebraic functions). The aim is to understand their geometry. Frequently techniques from several other fields are used like from algebraic topology, differential geometry or analysis, and the studies are tightly connected with these other branches of mathematics. This makes it nat- ural to work over the complex field C, even though other fields like function fields are important. To say that aims of algebraic geometry are totally geometric is half a lie (but a white one). The study of elliptic functions in the beginning of the 19th century, and subsequently of other algebraic functions, was the birth of modern algebraic geometry. The motivation and the origin was found in function theory, but the direction of research quickly took a geometric rout. Figure 1.1: The affine Riemann surfaces and algebraic curves appeared thogether with their function Fermat curve x50 ` y50 1. fields. “ The other main branch one could call “arithmetic”. Superficially presented, one studies numbers by geometric methods. An ultra famous example is Fermat’s last theorem, now Andrew Wiles’ theorem, that the equation xn ` yn zn has no integral solutions except the trivial ones. The arithmetic “ branch also relies on techniques from other fields, like number theory, Galois theory and representation theory. One very commonly applied technique is reduction modulo a prime number p. Hence the importance of including fields of positive characteristic among the base fields. Of course another very natural base field for many of these “arithmetic” studies is the field Q of Figure 1.2: The affine algebraic numbers. Fermat curve x51 ` Algebraic geometry is to the common benefit a triple marriage of geometry, y51 1. “ algebra and arithmetic. All of the spouses claim influence on the development 8notesforma4210— algebraic geometry i of the field which makes the field quit abstract; but also a most beautiful part of mathematics. 1.1 Fields and the affine space 1.1 We shall almost exclusively work over an algebraic closed field which we shall denote by k. In general we do not impose further constraints on k, except for a few results that require the characteristic to be zero. A specific field to have in mind would be the field of complex numbers C, but as indicated above, other important fields are Q and Fp. 1.2 The affine space An is just the space kn, but the name-change is there to underline that there is more to An than merely being a vector space— and hopefully, this will emerge from the fog during the course. Anyhow, in the beginning think about An as kn. Often the ground field will be tacitly understood, but when wanting to be precise about it, we shall write An k . p q The ground will always be algebraically closed unless the contrary is explicitly Figure 1.3: A one sheeted-hyperboloid. stated. Coordinates are not God-given but certainly man-made. So they are prone to being changed. General coordinate changes in An can be subtle, but trans- lation of the origin and linear changes are unproblematic, and will be done unscrupulously. They are called affine coordinate changes and the affine spaces An are named after them. 1.2 Closed algebraic sets The first objects we shall meet are the so called closed algebraic sets, and master Closed algebraic sets students in mathematics have already seen a great many examples of such. (lukkede algebraiske mengder) They are just subsets of the affine space An given by a certain number of polynomial equations. You have probably working with curves in the plane and may be with some surfaces in the space—like conic sections, hyperboloids and paraboloids, for example. 1.3 Formally the definition of a closed algebraic set is as follows. If S is a a, Ab, Bc, Cd, De, E f, Fg, Gh, Hi, Ij, J subset of the polynomial ring k x ,...,x , one defines r 1 ns k, Kl, Lm, Mn, No, O p, Pq, Qr, Rs, St, T Z S x An f x 0 for all f S , u, Uv, Vw, Wx, Xy, Y p q“t P | p q“ P u z, Z and subsets of An obtained in that way are the closed algebraic sets. Notice Mathematicians are that any linear combination of polynomials from S also vanishes at points always in shortage of of Z S , even if polynomials are allowed as coefficients. Therefore the ideal symbols and use all p q kinds of alphabets. The a generated by S has the same zero set as S; that is, Z S Z a . We shall p q“ p q germanic gothic letters almost exclusively work with ideals and tacitly replace a set of polynomials by are still in use in some the ideal it generates. context, like to denote ideals in some text. algebraic sets and the nullstellensatz 9 Any ideal in k x ,...,x is finitely generated, this is what Hilbert’s basis r 1 ns theorem tells us, so that a closed algebraic subset is described as the set of common zeros of finitely many polynomials. 10 notes for ma4210— algebraic geometry i Examples 1.1 The polynomial ring k x in one variable is a pid 1, so if a is an ideal in 1 A ring is a pid or a r s k x , it holds that a f x . Because polynomials in one variable merely principal ideal domain if it is an integral domain r s “pp qq 1 have finitely many zeros, the closed algebraic subsets of A are just the finite where every ideal is subsets of A1. principal 1.2 A more spectacular example is the so called Clebsch diagonal cubic; a surface in A3 C with equation p q x3 y3 z3 1 x y z 1 3. ` ` ` “p ` ` ` q An old plaster model of its reals points; that is, the points in A3 R satisfying p q the equation, is depicted in the margin. 1.3 The traditional conic sections are closed algebraic sets in A2.Aparabola is given as the zeros of y x2 and a hyperbola as the zeros of xy 1. ´ ´ K 1.4 The more constraints one imposes the smaller the solutions set will be, so if b a are two ideals, one has Z a Z b . The sum a b of two ideals has the Ñ p qÑ p q ` intersection Z a Z b as zero set; remembering that The Clebsch diagonal p qX p q cubic a b f g f a and g b ` “t ` | P P u one easily convinces oneself of this.
Load more

Recommended publications
  • The Geometry of Syzygies
    The Geometry of Syzygies A second course in Commutative Algebra and Algebraic Geometry David Eisenbud University of California, Berkeley with the collaboration of Freddy Bonnin, Clement´ Caubel and Hel´ ene` Maugendre For a current version of this manuscript-in-progress, see www.msri.org/people/staff/de/ready.pdf Copyright David Eisenbud, 2002 ii Contents 0 Preface: Algebra and Geometry xi 0A What are syzygies? . xii 0B The Geometric Content of Syzygies . xiii 0C What does it mean to solve linear equations? . xiv 0D Experiment and Computation . xvi 0E What’s In This Book? . xvii 0F Prerequisites . xix 0G How did this book come about? . xix 0H Other Books . 1 0I Thanks . 1 0J Notation . 1 1 Free resolutions and Hilbert functions 3 1A Hilbert’s contributions . 3 1A.1 The generation of invariants . 3 1A.2 The study of syzygies . 5 1A.3 The Hilbert function becomes polynomial . 7 iii iv CONTENTS 1B Minimal free resolutions . 8 1B.1 Describing resolutions: Betti diagrams . 11 1B.2 Properties of the graded Betti numbers . 12 1B.3 The information in the Hilbert function . 13 1C Exercises . 14 2 First Examples of Free Resolutions 19 2A Monomial ideals and simplicial complexes . 19 2A.1 Syzygies of monomial ideals . 23 2A.2 Examples . 25 2A.3 Bounds on Betti numbers and proof of Hilbert’s Syzygy Theorem . 26 2B Geometry from syzygies: seven points in P3 .......... 29 2B.1 The Hilbert polynomial and function. 29 2B.2 . and other information in the resolution . 31 2C Exercises . 34 3 Points in P2 39 3A The ideal of a finite set of points .
    [Show full text]
  • Hyperbolic Monopoles and Rational Normal Curves
    HYPERBOLIC MONOPOLES AND RATIONAL NORMAL CURVES Nigel Hitchin (Oxford) Edinburgh April 20th 2009 204 Research Notes A NOTE ON THE TANGENTS OF A TWISTED CUBIC B Y M. F. ATIYAH Communicated by J. A. TODD Received 8 May 1951 1. Consider a rational normal cubic C3. In the Klein representation of the lines of $3 by points of a quadric Q in Ss, the tangents of C3 are represented by the points of a rational normal quartic O4. It is the object of this note to examine some of the consequences of this correspondence, in terms of the geometry associated with the two curves. 2. C4 lies on a Veronese surface V, which represents the congruence of chords of O3(l). Also C4 determines a 4-space 2 meeting D. in Qx, say; and since the surface of tangents of O3 is a developable, consecutive tangents intersect, and therefore the tangents to C4 lie on Q, and so on £lv Hence Qx, containing the sextic surface of tangents to C4, must be the quadric threefold / associated with C4, i.e. the quadric determining the same polarity as C4 (2). We note also that the tangents to C4 correspond in #3 to the plane pencils with vertices on O3, and lying in the corresponding osculating planes. 3. We shall prove that the surface U, which is the locus of points of intersection of pairs of osculating planes of C4, is the projection of the Veronese surface V from L, the pole of 2, on to 2. Let P denote a point of C3, and t, n the tangent line and osculating plane at P, and let T, T, w denote the same for the corresponding point of C4.
    [Show full text]
  • Asymptotic Behavior of the Length of Local Cohomology
    ASYMPTOTIC BEHAVIOR OF THE LENGTH OF LOCAL COHOMOLOGY STEVEN DALE CUTKOSKY, HUY TAI` HA,` HEMA SRINIVASAN, AND EMANOIL THEODORESCU Abstract. Let k be a field of characteristic 0, R = k[x1, . , xd] be a polynomial ring, and m its maximal homogeneous ideal. Let I ⊂ R be a homogeneous ideal in R. In this paper, we show that λ(H0 (R/In)) λ(Extd (R/In,R(−d))) lim m = lim R n→∞ nd n→∞ nd e(I) always exists. This limit has been shown to be for m-primary ideals I in a local Cohen Macaulay d! ring [Ki, Th, Th2], where e(I) denotes the multiplicity of I. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extention modules may not have polynomial growth. Introduction Let R = k[x1, . , xd] be a polynomial ring over a field k, with graded maximal ideal m, and I ⊂ R d n a proper homogeneous ideal. We investigate the asymptotic growth of λ(ExtR(R/I ,R)) as a function of n. When R is a local Gorenstein ring and I is an m-primary ideal, then this is easily seen to be equal to λ(R/In) and hence is a polynomial in n. A theorem of Theodorescu and Kirby [Ki, Th, Th2] extends this to m-primary ideals in local Cohen Macaulay rings R. We consider homogeneous ideals in a polynomial ring which are not m-primary and show that a limit exists asymptotically although it can be irrational.
    [Show full text]
  • Algebraic Curves and Surfaces
    Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces .
    [Show full text]
  • Arxiv:1703.06832V3 [Math.AC]
    REGULARITY OF FI-MODULES AND LOCAL COHOMOLOGY ROHIT NAGPAL, STEVEN V SAM, AND ANDREW SNOWDEN Abstract. We resolve a conjecture of Ramos and Li that relates the regularity of an FI- module to its local cohomology groups. This is an analogue of the familiar relationship between regularity and local cohomology in commutative algebra. 1. Introduction Let S be a standard-graded polynomial ring in finitely many variables over a field k, and let M be a non-zero finitely generated graded S-module. It is a classical fact in commutative algebra that the following two quantities are equal (see [Ei, §4B]): S • The minimum integer α such that Tori (M, k) is supported in degrees ≤ α + i for all i. i • The minimum integer β such that Hm(M) is supported in degrees ≤ β − i for all i. i Here Hm is local cohomology at the irrelevant ideal m. The quantity α = β is called the (Castelnuovo–Mumford) regularity of M, and is one of the most important numerical invariants of M. In this paper, we establish the analog of the α = β identity for FI-modules. To state our result precisely, we must recall some definitions. Let FI be the category of finite sets and injections. Fix a commutative noetherian ring k. An FI-module over k is a functor from FI to the category of k-modules. We write ModFI for the category of FI-modules. We refer to [CEF] for a general introduction to FI-modules. Let M be an FI-module. Define Tor0(M) to be the FI-module that assigns to S the quotient of M(S) by the sum of the images of the M(T ), as T varies over all proper subsets of S.
    [Show full text]
  • 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, .
    [Show full text]
  • Hyperelliptic Curves with Many Automorphisms
    Hyperelliptic Curves with Many Automorphisms Nicolas M¨uller Richard Pink Department of Mathematics Department of Mathematics ETH Z¨urich ETH Z¨urich 8092 Z¨urich 8092 Z¨urich Switzerland Switzerland [email protected] [email protected] November 17, 2017 To Frans Oort Abstract We determine all complex hyperelliptic curves with many automorphisms and decide which of their jacobians have complex multiplication. arXiv:1711.06599v1 [math.AG] 17 Nov 2017 MSC classification: 14H45 (14H37, 14K22) 1 1 Introduction Let X be a smooth connected projective algebraic curve of genus g > 2 over the field of complex numbers. Following Rauch [17] and Wolfart [21] we say that X has many automorphisms if it cannot be deformed non-trivially together with its automorphism group. Given his life-long interest in special points on moduli spaces, Frans Oort [15, Question 5.18.(1)] asked whether the point in the moduli space of curves associated to a curve X with many automorphisms is special, i.e., whether the jacobian of X has complex multiplication. Here we say that an abelian variety A has complex multiplication over a field K if ◦ EndK(A) contains a commutative, semisimple Q-subalgebra of dimension 2 dim A. (This property is called “sufficiently many complex multiplications” in Chai, Conrad and Oort [6, Def. 1.3.1.2].) Wolfart [22] observed that the jacobian of a curve with many automorphisms does not generally have complex multiplication and answered Oort’s question for all g 6 4. In the present paper we answer Oort’s question for all hyperelliptic curves with many automorphisms.
    [Show full text]
  • 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.
    [Show full text]
  • 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 ...........................
    [Show full text]
  • Classical Algebraic Geometry
    CLASSICAL ALGEBRAIC GEOMETRY Daniel Plaumann Universität Konstanz Summer A brief inaccurate history of algebraic geometry - Projective geometry. Emergence of ’analytic’geometry with cartesian coordinates, as opposed to ’synthetic’(axiomatic) geometry in the style of Euclid. (Celebrities: Plücker, Hesse, Cayley) - Complex analytic geometry. Powerful new tools for the study of geo- metric problems over C.(Celebrities: Abel, Jacobi, Riemann) - Classical school. Perfected the use of existing tools without any ’dog- matic’approach. (Celebrities: Castelnuovo, Segre, Severi, M. Noether) - Algebraization. Development of modern algebraic foundations (’com- mutative ring theory’) for algebraic geometry. (Celebrities: Hilbert, E. Noether, Zariski) from Modern algebraic geometry. All-encompassing abstract frameworks (schemes, stacks), greatly widening the scope of algebraic geometry. (Celebrities: Weil, Serre, Grothendieck, Deligne, Mumford) from Computational algebraic geometry Symbolic computation and dis- crete methods, many new applications. (Celebrities: Buchberger) Literature Primary source [Ha] J. Harris, Algebraic Geometry: A first course. Springer GTM () Classical algebraic geometry [BCGB] M. C. Beltrametti, E. Carletti, D. Gallarati, G. Monti Bragadin. Lectures on Curves, Sur- faces and Projective Varieties. A classical view of algebraic geometry. EMS Textbooks (translated from Italian) () [Do] I. Dolgachev. Classical Algebraic Geometry. A modern view. Cambridge UP () Algorithmic algebraic geometry [CLO] D. Cox, J. Little, D.
    [Show full text]
  • Determining the Automorphism Group of a Hyperelliptic Curve
    Determining the Automorphism Group of a Hyperelliptic Curve ∗ Tanush Shaska Department of Mathematics University of California at Irvine 103 MSTB, Irvine, CA 92697 ABSTRACT 1. INTRODUCTION In this note we discuss techniques for determining the auto- Let Xg be an algebraic curve of genus g defined over an morphism group of a genus g hyperelliptic curve Xg defined algebraically closed field k of characteristic zero. We denote over an algebraically closed field k of characteristic zero. by Aut(Xg) the group of analytic (equivalently, algebraic) The first technique uses the classical GL2(k)-invariants of automorphisms of Xg. Then Aut(Xg) acts on the finite set binary forms. This is a practical method for curves of small of Weierstrass points of Xg. This action is faithful unless genus, but has limitations as the genus increases, due to the Xg is hyperelliptic, in which case its kernel is the group of fact that such invariants are not known for large genus. order 2 containing the hyperelliptic involution of Xg. Thus The second approach, which uses dihedral invariants of in any case, Aut(Xg) is a finite group. This was first proved hyperelliptic curves, is a very convenient method and works by Schwartz. In 1893 Hurwitz discovered what is now called well in all genera. First we define the normal decomposition the Riemann-Hurwitz formula. From this he derived that of a hyperelliptic curve with extra automorphisms. Then dihedral invariants are defined in terms of the coefficients of |Aut(Xg)| ≤ 84 (g − 1) this normal decomposition. We define such invariants inde- which is known as the Hurwitz bound.
    [Show full text]
  • 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.
    [Show full text]