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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

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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
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 . . . . . . . . . . . . . . . . . . . . . . . .

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