MATH 137 NOTES: UNDERGRADUATE ALGEBRAIC GEOMETRY AARON LANDESMAN CONTENTS 1. Introduction 6 2. Conventions 6 3. 1/25/16 7 3.1. Logistics 7 3.2. History of Algebraic Geometry 7 3.3. Ancient History 7 3.4. Twentieth Century 10 3.5. How the course will proceed 10 3.6. Beginning of the mathematical portion of the course 11 4. 1/27/16 13 4.1. Logistics and Review 13 5. 1/29/16 17 5.1. Logistics and review 17 5.2. Twisted Cubics 17 5.3. Basic Definitions 21 5.4. Regular functions 21 6. 2/1/16 22 6.1. Logistics and Review 22 6.2. The Category of Affine Varieties 22 6.3. The Category of Quasi-Projective Varieties 24 7. 2/4/16 25 7.1. Logistics and review 25 7.2. More on Regular Maps 26 7.3. Veronese Maps 27 7.4. Segre Maps 29 8. 2/5/16 30 8.1. Overview and Review 30 8.2. More on the Veronese Variety 30 8.3. More on the Segre Map 32 9. 2/8/16 33 9.1. Review 33 1 2 AARON LANDESMAN 9.2. Even More on Segre Varieties 34 9.3. Cones 35 9.4. Cones and Quadrics 37 10. 2/10/15 38 10.1. Review 38 10.2. More on Quadrics 38 10.3. Projection Away from a Point 39 10.4. Resultants 40 11. 2/12/16 42 11.1. Logistics 42 11.2. Review 43 11.3. Projection of a Variety is a Variety 44 11.4. Any cone over a variety is a variety 44 11.5. Projection of a Variety from a Product is a Variety 45 11.6. The image of a regular map is a Variety 45 12. 2/17/16 46 12.1. Overview 46 12.2. Families 47 12.3. Universal Families 47 12.4. Sections of Universal Families 49 12.5. More examples of Families 50 13. 2/19/16 51 13.1. Review 51 13.2. Generality 51 13.3. Introduction to the Nullstellensatz 54 14. 2/22/16 55 14.1. Review and Overview of Coming Attractions 55 14.2. What scissors are good for: Cutting out 56 14.3. Irreducibility 58 15. 2/24/16 59 15.1. Overview 59 15.2. Preliminaries 59 15.3. Primary Decomposition 60 15.4. The Nullstellensatz 61 16. 2/26/15 63 16.1. Review of the Nullstellensatz 63 16.2. Preliminaries to Completing the proof of the Nullstellensatz 63 16.3. Proof of Nullstellensatz 64 16.4. Grassmannians 65 17. 2/29/16 66 17.1. Overview and a homework problem 66 MATH 137 NOTES: UNDERGRADUATE ALGEBRAIC GEOMETRY 3 17.2. Multilinear algebra, in the service of Grassmannians 68 17.3. Grassmannians 70 18. 3/2/16 70 18.1. Review 70 18.2. The grassmannian is a projective variety 71 18.3. Playing with grassmannian 73 19. 3/4/16 74 19.1. Review of last time: 74 19.2. Finding the equations of G(1, 3) 75 19.3. Subvarieties of G(1, 3) 77 19.4. Answering our enumerative question 78 20. 3/7/16 79 20.1. Plan 79 20.2. Rational functions 79 21. 3/9/16 82 21.1. Rational Maps 82 21.2. Operations with rational maps 82 22. 3/11/16 85 22.1. Calculus 85 22.2. Blow up of P2 at a point 86 22.3. Blow ups in general 87 22.4. Example: join of a variety 88 23. 3/21/16 88 23.1. Second half of the course 88 23.2. Dimension historically 89 23.3. Useful characterization of dimension 90 23.4. The official definition 91 23.5. Non-irreducible, non-projective case 91 24. 3/23/16 92 24.1. Homework 92 24.2. Calculating dimension 92 24.3. Dimension of the Grassmannian 94 24.4. Dimension of the universal k-plane 94 24.5. Dimension of the variety of incident planes 95 25. 3/25/16 96 25.1. Review 96 25.2. Incidence varieties 97 25.3. Answering Question 25.1 98 26. 3/28/16 99 26.1. Secant Varieties 99 26.2. Deficient Varieties 100 27. 3/30/16 103 4 AARON LANDESMAN 27.1. Schedule 103 27.2. The locus of matrices of a given rank 104 27.3. Polynomials as determinants 106 28. 4/1/16 108 28.1. Proving the main theorem of dimension theory 108 28.2. Fun with dimension counts 110 29. 4/4/16 113 29.1. Overview 113 29.2. Hilbert functions 113 30. 4/6/16 118 30.1. Overview and review 118 30.2. An alternate proof of Theorem 30.1, using the Hilbert syzygy theorem 119 31. 3/8/16 121 31.1. Minimal resolution of the twisted cubic 121 31.2. Tangent spaces and smoothness 124 32. 4/11/16 126 32.1. Overview 126 32.2. Definitions of tangent spaces 126 32.3. Constructions with the tangent space 128 33. 4/13/16 130 33.1. Overview 130 33.2. Dual Varieties 130 33.3. Resolution of Singularities 132 33.4. Nash Blow Ups 132 34. 4/15/16 134 34.1. Bertini’s Theorem 134 34.2. The Lefschetz principle 135 34.3. Degree 137 35. 4/18/16 139 35.1. Review 139 35.2. Bezout’s Theorem, part I: A simpler statement 140 35.3. Examples of Bezout, part I 141 36. 4/20/16 143 36.1. Overview 143 36.2. More degree calculations 143 37. Proof of Bezout’s Theorem 145 38. 4/22/16 147 38.1. Overview 147 38.2. Binomial identities 147 38.3. Review of the setup from the previous class 148 38.4. Proving Bezout’s theorem 149 MATH 137 NOTES: UNDERGRADUATE ALGEBRAIC GEOMETRY 5 38.5. Strong Bezout 150 39. 4/25/16 152 39.1. Overview 152 39.2. The question of connected components over the reals 153 39.3. The genus is constant in families 153 39.4. Finding the genus 155 39.5. Finding the number of connected components of R 156 40. 4/27/16 158 40.1. Finishing up with the number of connected components of real plane curves 158 40.2. Parameter spaces 158 40.3. Successful Attempt 4 161 6 AARON LANDESMAN 1. INTRODUCTION Joe Harris taught a course (Math 137) on undergraduate algebraic geometry at Harvard in Spring 2016. 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 philo- sophical 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 Several of the classes have notes taken by Hannah Larson. Thanks to her for taking notes when I missed class. Please email suggestions to [email protected]. 2. CONVENTIONS Here are some conventions we will adapt throughout the notes. (1) I have a preference toward making any detail stated in class, which is not verified, into an exercise. This will mean there are many trivial exercises in the notes. When the exercise seems nontrivial, I will try to give a hint. Feel free to contact me if there are any exercises you do not know how to solve or other details which are unclear. (2) Throughout the notes, I will often include parenthetical re- marks which describe things beyond the scope of the course. If some word or explanation is placed in quotation marks or in parentheses, and you don’t understand it, don’t worry about it! It’s more meant to give you a flavor of how one might describe the 19th century ideas in this course in terms of 20th century algebraic geometry. 1This introduction has been adapted from Akhil Matthew’s introduction to his notes, with his permission. MATH 137 NOTES: UNDERGRADUATE ALGEBRAIC GEOMETRY 7 3. 1/25/16 3.1. Logistics. (1) Three lectures a week on Monday, Wednesday, and Friday (2) We’ll have weekly recitations (3) This week we’ll have special recitations on Wednesday and Friday at 3pm; (4) You’re welcome to go to either section (5) Aaron will be taking notes (6) Please give feedback to Professor Harris or the CAs (7) We’ll have weekly problem sets, due Fridays. The first prob- lem set might be due next Wednesday or so. (8) The minimal prerequisite is Math 122. (9) The main text is Joe Harris’ “A first course in algebraic geom- etry.” (10) The book goes fast in many respects. In particular, the defini- tion of projective space is given little attention. (11) Some parts of other math will be used. For example, knowing about topology or complex analysis will be useful to know, but we’ll define every term we use. 3.2. History of Algebraic Geometry. Today, we will go through the History of algebraic geometry. 3.3. Ancient History. The origins of algebraic geometry are in un- derstanding solutions of polynomial equations. People first stud- ied this around the late middle ages and early renaissance. People started by looking at a polynomial in one variables f(x). You might then want to know: Question 3.1. What are the solutions to f(x) = 0? (1) If f(x) is a quadratic polynomial, you can use the quadratic formula.