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Geometry of Algebraic Curves

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Geometry of Algebraic Curves Geometry of Algebraic Curves Lectures delivered by Joe Harris Notes by Akhil Mathew Fall 2011, Harvard Contents Lecture 1 9/2 x1 Introduction 5 x2 Topics 5 x3 Basics 6 x4 Homework 11 Lecture 2 9/7 x1 Riemann surfaces associated to a polynomial 11 x2 IOUs from last time: the degree of KX , the Riemann-Hurwitz relation 13 x3 Maps to projective space 15 x4 Trefoils 16 Lecture 3 9/9 x1 The criterion for very ampleness 17 x2 Hyperelliptic curves 18 x3 Properties of projective varieties 19 x4 The adjunction formula 20 x5 Starting the course proper 21 Lecture 4 9/12 x1 Motivation 23 x2 A really horrible answer 24 x3 Plane curves birational to a given curve 25 x4 Statement of the result 26 Lecture 5 9/16 x1 Homework 27 x2 Abel's theorem 27 x3 Consequences of Abel's theorem 29 x4 Curves of genus one 31 x5 Genus two, beginnings 32 Lecture 6 9/21 x1 Differentials on smooth plane curves 34 x2 The more general problem 36 x3 Differentials on general curves 37 x4 Finding L(D) on a general curve 39 Lecture 7 9/23 x1 More on L(D) 40 x2 Riemann-Roch 41 x3 Sheaf cohomology 43 Lecture 8 9/28 x1 Divisors for g = 3; hyperelliptic curves 46 x2 g = 4 48 x3 g = 5 50 1 Lecture 9 9/30 x1 Low genus examples 51 x2 The Hurwitz bound 52 2.1 Step 1 . 53 2.2 Step 10 ................................. 54 2.3 Step 100 ................................ 54 2.4 Step 2 . 54 Lecture 10 10/7 x1 Preliminary remarks 57 x2 The next theorem 57 x3 Remarks 58 x4 The main result 59 Lecture 11 10/12 x1 Recap 62 x2 The bound, again 64 x3 The general position lemma 64 x4 Monodromy 65 x5 Proof of the general position lemma 66 Lecture 12 10/14 x1 General position lemma 68 x2 Projective normality 69 x3 Sharpness of the Castelnuovo bound 70 x4 Scrolls 71 Lecture 13 10/17 x1 Motivation 74 x2 Scrolls again 75 x3 Intersection numbers 75 Lecture 14 10/19 x1 A basic lemma 79 x2 Castelnuovo's lemma 81 x3 Equality in the Castelnuovo bound 82 Lecture 15 10/21 x1 Introduction 84 x2 Castelnuovo's argument in the second case 85 x3 Converses 87 Lecture 16 10/26 x1 Goals 89 x2 Inflection points 90 x3 A modern reformulation 91 Lecture 17 11/2 x1 Review 95 x2 The Gauss map 96 x3 Plane curves 97 Lecture 18 11/4 x1 Inflectionary points; one last remark 101 x2 Weierstrass points 102 x3 Examples 105 Lecture 19 11/9 x1 Real algebraic curves 106 x2 Singularities 108 x3 Harnack's theorem 108 x4 Nesting 109 x5 Proof of Harnack's theorem 110 Lecture 20 11/16 x1 Basic notions 111 x2 The differential of u 112 x3 Marten's theorem 115 Lecture 21 11/18 x1 Setting things up again 116 x2 Theorems 118 Lecture 22 11/30 x1 Recap 121 x2 Families 122 x3 The basic construction 123 x4 Constructing such families 124 x5 Specializing linear series 125 Lecture 23 12/2 x1 Two special cases 127 x2 The inductive step 128 x3 The basic construction 128 x4 A relation between the ramifications of U; W 130 x5 Finishing 131 Introduction Joe Harris taught a course (Math 287y) on the geometry of algebraic curves at Harvard in Fall 2011. These are my \live-TEXed" notes from the course. Conventions are as follows: Each lecture gets its own \chapter," and appears in the table of contents with the date. Some lectures are marked \section," which means that they were taken at a recitation session. The recitation sessions were taught by Anand Deopurkar. Of course, these notes are not a faithful representation of the course, either in the mathematics itself or in the quotes, jokes, and philosophical musings; in particular, the errors are my fault. By the same token, any virtues in the notes are to be credited to the lecturer and not the scribe. Please email corrections to [email protected]. Lecture 1 Geometry of Algebraic Curves notes Lecture 1 9/2 x1 Introduction The text for this course is volume 1 of Arborello-Cornalba-Griffiths-Harris, which is even more expensive nowadays. We will be covering a subset of the book, and probably adding some additional topics, but this will be the basic source for most of the stuff we do. There will be weekly homeworks, and that will determine the grade for those of you taking the course for a grade. There will be no final. There will be a weekly section with Anand Deopurkar. The course doesn't meet on Mondays because of the \Basic Notions" seminar. On those weeks that there won't be seminars, we might meet then. Every third or fourth week, the lecture will be cut short on Wednesdays. However, the course will basically meet on a three-hour-per-week basis. x2 Topics Here is what we are going to talk about. We are going to talk about compact Riemann surfaces, and a compact Riemann surface is the same thing as a smooth projective algebraic curve (over C). That in turn is really the same thing as a smooth projective curve over any algebraically closed field of characteristic zero. By abuse of notation, we will use C to denote any such field as well. The fact that these are the same thing|that is, that a compact Riemann surface is an algebraic curve|is nontrivial. It requires work to show that such an object even admits a nontrivial meromorphic function. Note: 1.1 Proposition. There are compact complex manifolds of dim ≥ 2 that do not admit nonconstant meromorphic functions. The miracle in dimension one is that there are nonconstant meromorphic functions, and enough to embed the manifold in projective space. There are a number of beautiful topics that we are not going to cover. We will not talk about singular algebraic curves, in general. We will encounter them (e.g. when we consider maps of curves to projective space, the image might be singular), but they will not be the focus of study. When we have a singular curve C in projective space, we will treat C its normalization, i.e. as an image of a smooth projective curves. We also will not talk about open (i.e. noncompact) Riemann surfaces. Another even- more large-scale and interesting topic is the theory of families of curves, and how the isomorphism class of a curve changes as we vary the coefficients of the defining equations (e.g. moduli of curves). Finally, we are not going to talk about curves over fields that are not algebraically closed. There has been a huge amount of work on algebraic curves over R, but we won't discuss them. 5 Lecture 1 Geometry of Algebraic Curves notes x3 Basics Today, we shall set the notation and conventions. Algebraic curves is one of the oldest subjects in modern mathematics, as it was one of the first things people did once they learned about polynomials. It has developed over time a multiplicity of language and symbols, and we will run through it. Let X be a smooth projective algebraic curve over C. There are many ways of defining the genus of X, e.g. via the Hilbert polynomial, the Euler characteristic (via coherent cohomology), and so on. We are just going to take the naive point of view. 1.2 Definition. The genus of X is the topological genus (as a surface). We can also use: 1. g(X) = 1 − χ(OX ). 1 2. 1 − 2 χtop(X). 1 3. 2 deg KX + 1 (for KX the canonical divisor, see below). Given a geometric object, one wishes to define the functions on it. On a compact Riemann surface, there are no nonconstant regular functions, by the maximal principle: every holomorphic function on X is constant. We need to allow poles, and to keep track of them we will need to introduce the language of divisors. 1.3 Definition. A divisor D on X is a formal finite linear combination of points P nipi; pi 2 X on the curve. We say that D is effective if all ni ≥ 0 (in which case P we write D ≥ 0), and we say that deg D = ni is the degree of D. As a reality check, one should see that the family of effective divisors of a given degree d should be the dth symmetric power of the curve with itself, Cd; this is C × · · · × C (d times) modulo the symmetric group Sd. 1 The point of this is, if we have a meromorphic function f : X ! P , we can associate to it a divisor measuring its zeros and poles. 1 1.4 Definition. Given f : X ! P , we say that the divisor of X is the sum P p2X ordp(f)p. The positive terms come from the zeros, while the negative terms come from the poles. The problem is to deal with interesting classes of functions. Holomorphic functions do it, while there are too many meromorphic functions to work with them all at once. Instead, we do the following: P 1.5 Definition. Given a divisor D = npp, we can look at functions which may have poles at each p, but with orders bounded by np. Namely, we look at the space L(D) 1 of rational functions f : X ! P such that ordp(f) ≥ −np for all p 2 X. This is equivalently the space of f such that div(f) + D ≥ 0.
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