Harvard-137.pdf - Math 137 Lecture Notes Evan Chen Spring...

This preview shows page 1 - 4 out of 66 pages.

Math 137 Lecture Notes Evan Chen Spring 2015 This is Harvard College’s Math 137 , instructed by Yaim Cooper. The formal name for this class is “Algebraic Geometry”; we will be studying complex varieties. The permanent URL for this document is coursework.html , along with all my other course notes. As usual, Dropbox links expire at the end of the semester. Contents 1 February 6, 2015 5 1.1 Facts about ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 C -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Hilbert Basis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Viewing varieties as ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 Flavors of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 February 11, 2015 8 2.1 A Small Remark from Evan o’Dorney (in response to homework) . . . . . 8 2.2 Obtaining Ideals From Varieties . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Obtaining Varieties from Ideal . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Maximal Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Hilbert’s Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 February 13, 2015 11 3.1 Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 February 18, 2015 12 4.1 A Coordinate-Change Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Proof of Weak Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.3 Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5 February 20, 2015 15 5.1 Coordinate Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.2 Pullback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.3 Which rings are coordinate rings? . . . . . . . . . . . . . . . . . . . . . . 16 5.4 The equivalence of algebra and geometry . . . . . . . . . . . . . . . . . . 16 6 February 23, 2015 18 6.1 Recap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 6.2 Isomorphism of Affine Algebraic Varieties . . . . . . . . . . . . . . . . . . 18 1
Image of page 1

Subscribe to view the full document.

Evan Chen (Spring 2015) Contents 6.3 (Digression) Spectrum of a Ring . . . . . . . . . . . . . . . . . . . . . . . 19 6.4 Complex Projective Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.5 Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 6.6 Projective Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 7 February 25, 2015 21 7.1 Functions on a Projective Variety . . . . . . . . . . . . . . . . . . . . . . . 21 7.2 Projective Analogues of Affine Results . . . . . . . . . . . . . . . . . . . . 22 7.3 Transforming affine varieties to projective ones . . . . . . . . . . . . . . . 22 8 February 27, 2015 24 8.1 Ideals of Projective Closures: A Cautionary Tale . . . . . . . . . . . . . . 24 8.2 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 8.3 Morphisms of projective varieties . . . . . . . . . . . . . . . . . . . . . . . 25 8.4 Examples of projective maps . . . . . . . . . . . . . . . . . . . . . . . . . 25 9 March 2, 2015 26 9.1 Examples of Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 9.2 Isomorphisms of Projective Varieties . . . . . . . . . . . . . . . . . . . . . 26 9.3 Projective Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 9.4 Quasi-projective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 10 March 4, 2015 29 10.1 Morphisms (and Examples) of Quasi-Projective Varieties . . . . . . . . . 29 10.2 Affine quasi-projective varieties . . . . . . . . . . . . . . . . . . . . . . . . 29 10.3 Rings of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 10.4 Complements of hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 30 11 March 6, 2015 32 11.1 Quasi-projective varieties are covered by locally affine sets. . . . . . . . . 32 11.2 Regular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 12 March 9, 2015 34 12.1 Basis of Open Affine Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 12.2 Regular Functions Continued . . . . . . . . . . . . . . . . . . . . . . . . . 34 12.3 Regular functions on quasi-projective varieties . . . . . . . . . . . . . . . 35 12.4 Recasting morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 13 March 11, 2015 36 13.1 Veronese Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 13.2 Ring of regular functions on projective spaces . . . . . . . . . . . . . . . . 37 14 March 23, 2015 39 14.1 More on Veronese Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 14.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 15 March 27, 2015 40 15.1 Review of Projective Closures . . . . . . . . . . . . . . . . . . . . . . . . . 40 15.2 Enumerative Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 16 March 30, 2015 42 16.1 Glimpses of Enumerative Geometry . . . . . . . . . . . . . . . . . . . . . 42 2
Image of page 2
Evan Chen (Spring 2015) Contents 16.2 Segre Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 17 April 1, 2015 44 17.1 Segre maps continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 17.2 Topology of CP m × CP n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 17.3 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 17.4 General Equations Cutting Out Σ n,m . . . . . . . . . . . . . . . . . . . . 45 18 April 3, 2015 46 18.1 Grassmanians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 19 April 6, 2015 48 19.1 Representation of Grassmanians as Matrices . . . . . . . . . . . . . . . . 48 19.2 Dimension of the Grassmanian . . . . . . . . . . . . . . . . . . . . . . . . 48 19.3 Embedding Grassmanian into Projective Space . . . . . . . . . . . . . . . 48 19.4 Grassmanian is a Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 20 April 8, 2015 50 20.1 Grassmanian is a Complex Manifold . . . . . . . . . . . . . . . . . . . . . 50 20.2 Degree of a Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 20.3 Degree is not preserved under isomorphism! . . . . . . . . . . . . . . . . . 50 21 April 10, 2015 52 21.1 Degree of the Veronese Map . . . . . . . . . . . . . . . . . . . . . . . . . . 52 21.2 Complete Intersections and Degrees . . . . . . . . . . . . . . . . . . . . . 52 21.3 Curves in Projective Plane . . . . . . . . . . . . . . . . . . . . . . . . . . 53 21.4 How do you tell apart curves? . . . . . . . . . . . . . . . . . . . . . . . . . 53 22 April 13, 2015 54 22.1 Tangent Space of Affine Spaces . . . . . . . . . . . . . . . . . . . . . . . .
Image of page 3

Subscribe to view the full document.

Image of page 4
  • Fall '19

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes