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Notes for Math 282, Geometry of Algebraic Curves

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Notes for Math 282, Geometry of Algebraic Curves NOTES FOR MATH 282, GEOMETRY OF ALGEBRAIC CURVES AARON LANDESMAN CONTENTS 1. Introduction 4 2. 9/2/15 5 2.1. Course Mechanics and Background 5 2.2. The Basics of curves, September 2 5 3. 9/4/15 6 3.1. Outline of Course 6 4. 9/9/15 8 4.1. Riemann Hurwitz 8 4.2. Equivalent characterizations of genus 8 4.3. Consequences of Riemann Roch 9 5. 9/11/15 10 6. 9/14/15 11 6.1. Curves of low genus 12 7. 9/16/15 13 7.1. Geometric Riemann-Roch 13 7.2. Applications of Geometric Riemann-Roch 14 8. 9/18/15 15 8.1. Introduction to parameter spaces 15 8.2. Moduli Spaces 16 9. 9/21/15 18 9.1. Hyperelliptic curves 18 10. 9/23/15 20 10.1. Hyperelliptic Curves 20 10.2. Gonal Curves 21 11. 9/25/15 22 11.1. Curves of genus 5 23 12. 9/28/15 24 12.1. Canonical curves of genus 5 24 12.2. Adjoint linear series 25 13. 9/30/15 27 13.1. Program for the remainder of the semester 27 13.2. Adjoint linear series 27 14. 10/2/15 29 15. 10/5/15 32 15.1. Castelnuovo’s Theorem 33 16. 10/7/15 34 17. 10/9/15 36 18. 10/14/15 38 1 2 AARON LANDESMAN 19. 10/16/15 41 19.1. Review 41 19.2. Minimal Varieties 42 20. 10/19/15 44 21. 10/21/15 47 22. 10/23/15 49 22.1. Agenda and Review 49 22.2. Resolutions of Projective Varieties and Green’s conjecture 50 22.3. The Maximal Rank Conjecture 52 23. 10/26/15 53 24. 10/28/15 55 24.1. Review 55 25. 10/30/15 57 25.1. Review 57 25.2. Today: Brill Noether Theorem in dimension at least 3 58 26. 11/2/15 61 26.1. Review 61 26.2. Hilbert Schemes 61 27. 11/4/15 64 27.1. Logistics 64 27.2. Tangent Spaces in Brill Noether Theory 64 27.3. Martens Theorem 66 28. 11/6/15 67 29. 10/9/15 71 29.1. Agenda and Review 71 30. 11/11/15 75 30.1. Review 75 30.2. The Genus 6 Canonical Model 76 31. 11/13/15 78 32. 11/16/15 81 32.1. Logistics and Review 81 32.2. A continuing study of genus 6 curves 82 33. 11/18/15 84 33.1. Logistics and Overview 84 34. 11/20/15 87 34.1. Plan, conventions, and review. 87 34.2. An Upper bound on the dimension 88 34.3. Proof of Existence for Brill Noether 88 35. 11/23/15 90 35.1. Review 90 35.2. Inflectionary points of linear series 92 36. 11/30/15 93 36.1. Review and overview 93 37. 12/2/15 97 37.1. Overview 97 37.2. Retraction 97 37.3. Finishing the proof of Brill-Noether 97 NOTES FOR MATH 282, GEOMETRY OF ALGEBRAIC CURVES 3 37.4. Further Questions 99 4 AARON LANDESMAN 1. INTRODUCTION Joe Harris taught a course (Math 282) on algebraic curves at Harvard in Fall 2015. 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. 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. 1 Please email corrections to [email protected]. 1This introduction has been adapted from Akhil Matthew’s introduction to his notes, with his permission. NOTES FOR MATH 282, GEOMETRY OF ALGEBRAIC CURVES 5 2. 9/2/15 2.1. Course Mechanics and Background. (1) Math 282, Algebraic Curves (2) CA Adrian (3) Text: ACGH, Volume 1 (4) Four years ago, a similar course was taught, following ACGH. The idea was: given a curve, what can we say about it. This is only half the story. Curves can appear in the abstract and in projective space. An important part of understanding curves is how they vary in flat families. The differ- ence between ACGH volumes 1 and 2, is that 1 deals with a fixed curve and 2 deals with families of curves. To learn more on families of curves, look at Moduli of Curves. (5) Two major changes in the language since when the book was written: First, we will use cohomology, and second we will use schemes. (6) Algebraic curves were first studied over the complex numbers. Some peo- ple studied complex analysis of Riemann Surfaces, and others studied polynomials in two variables. Remark 2.1. We will use the language of smooth projective curves and compact Riemann surfaces interchangeably. We will assume all curves are over the complex numbers. The central problem of the course is Question 2.2. What is a curve? In the 19th century, a curve is a subset of Pn for some n. In the 20th century, a curve became an abstract curve, which exists indepen- dently of any particular embedding in projective space. A similar perspective was adapted in group theory. Originally, people viewed groups as subsets of GLn. Now, this is called representation theory. Remark 2.3. In his textbook, Hartshorne says the goal of algebraic geometry is to classify algebraic varieties. In the modern context, we can just specify the genus. However, in the 19th century, you would have to also specify the degree. We can then ask, which pairs of d, g are realized as a curve. This is still not completely known. You can also specify a projective space, and then ask which curves can be realized in that projective space. 2.2. The Basics of curves, September 2. 1 Definition 2.4. Define the genus by g = 2 (1 - χtopX). Definition 2.5. An ordinary singular point of a curve of multiplicity m is a point in which m branches of a curve meet transversely of a point. We can define this more rigorously by saying that the completion of its local ring is isomorphic to k[x, y]/(f1 ··· fm) where fi are distinct linear functions in x, y. Lemma 2.6. Let X be a curve. The following are equivalent: (1) The genus of X. (2) 1 - χOX 1 (3) 2 (deg KX + 2) 6 AARON LANDESMAN (4) 1 - c, where c is the constant term of the Hilbert polynomial of C ⊂ Pr. ∼ 2 (5) If C = C0 ⊂ P of degree d with ordinary singular points of multiplicity mi, d-1 mi then g(C) = 2 - i 2 Definition 2.7. A divisor isP an formal sum of the form D = i nipi for ni 2 Z, pi 2 X. We say D is effective if D ≥ 0. We define the degree by deg D = i ni. For f a rational function on X, we define P P div f = (f) = ordp(f) · p = (f)0 -(f) p X . 1 Remark 2.8. By the residue formula applied to the logarithmic derivative of a rational function f, we have deg (f) = 0. Definition 2.9. We say D ∼ E, or D is linearly equivalent to E if there exists a rational function f with (f) = D - E. Effective divisors of degree d on X will be d notated as Cd, which is by definition C /Sd. Definition 2.10. Given D 2 div X, we look at L(D) := ff 2 K(X)× :(f) + D ≥ 0g. An alternative notation for L is H0. Since we can specify the polar part of such a function, this is a finite dimensional vector space. This uses the fact that there are no nonconstant meromorphic function. We define `(D) = dim L(D) and define r(D) = `(D)- 1. Remark 2.11. If D ∼ E then L(D) =∼ L(E), as given by multiplication by f. Definition 2.12. We define Picd(X) := Divisors of degree d/ ∼. Remark 2.13. It turns out this set corresponds to the points of a variety. Definition 2.14. Suppose ! is a rational 1-form, which looks locally like f(z) dz. 2g-2 We let (!) = ordp(!)p. We define KX := (w) 2 Pic (X). P 3. 9/4/15 Note: there will be no class Monday or Friday. 3.1. Outline of Course. (1) This week: Basics of linear series (2) Starting 9/14, we’ll talk about curves of low genus and Castelnuovo’s the- orem (gives an upper bound on the genus of a curve of degree d in projec- tive space) (3) Brill-Noether Theory: This addresses the question “What can you say about a general curve?” It makes sense to ask whether there is an open subset of the Hilbert scheme on which there is a uniformity of the corresponding curves. Remark 3.1. For the remainder of the course, we let X be a smooth projective curve. We let D = i niPi and say D ∼ E if D is linearly equivalent to E. We let KX be the divisor class of the canonical sheaf !, of degree 2g - 2. Use L(D) for P H0(C, D), let `(D) = h0(C, D) and r(D) = `(D)- 1. NOTES FOR MATH 282, GEOMETRY OF ALGEBRAIC CURVES 7 The justification of looking at these linear systems is that we only allow poles at specified points.
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