Springer UTX, 2000

Algebraic Geometry

"The aim of this book is to describe the underlying principles of algebraic geometry, some of its important developments in the twentieth century, and some of the problems that occupy its practitioners today."

Chapter 1 : Affine Algebraic VarietiesEdit

Section 1.1 : Definition and ExamplesEdit

Section 1.2 : The Zariski TopologyEdit

Section 1.3 : Morphisms of Affine Algebraic VarietiesEdit

Section 1.4 : DimensionEdit

Chapter 2 : Algebraic FoundationsEdit

Chapter 3 : Projective VarietiesEdit

Section 3.1 : Projective SpaceEdit

Section 3.2 : Projective VarietiesEdit

Section 3.3 : The Projective Closure of an Algebraic VarietyEdit

  • The closure $ \overline{V} $ of the twisted cubic: $ [1:t:t^2:t^3] = \left[\frac{1}{t^3}:\frac{1}{t^2}:\frac{1}{t}:1\right] \rightarrow [0:0:0:1] $ as $ t \rightarrow \infty $

Section 3.4 : Morphisms of Projective VarietiesEdit

  • no single choice of polynomials will work for all points of C: is this supposed to be immediately clear?
  • For example, the point at infinity is [0:0:1] when we consider the chart $ U_x $: At first it seemed like there should be lots of points at infinity, but then I realized that the points on the "line at infinity" don't count: $ \mathbb{P}^2 = U_x \sqcup (\text{the line at infinity}) \sqcup (\text{the point at infinity}) $
  • A "form" is a homogeneous polynomial (in some number of variables).
  • Exercise 3.4.4: $ \mathbb{V}(x) $ can be parametrized as [p:0:q:r], and $ \mathbb{V}(x-y^4-z^4) $ can be parametrized as $ \left[s^4:t^4+u^4:ts^3:us^3\right] $

Section 3.5 : Automorphisms of Projective SpaceEdit

  • Exercise 3.5.1: The 3 is very important here! This does not hold in other dimensions.

Chapter 4 : Quasi-Projective VarietiesEdit

Chapter 5 : Classical ConstructionsEdit

Section 5.1 : Veronese MapsEdit

Section 5.2 : Five Points Determine a ConicEdit

Section 5.3 : The Segre Map and Products of VarietiesEdit

Section 5.4 : GrassmanniansEdit

  • as the action of g on the determinants is just multiplication by the nonzero constant deg g: clearly k x k subsets must be special somehow... ah - just delete the irrelevant rows of $ (a_{ij}) $ and work in $ {\rm GL}_k(\mathbb{C}) $. (The rows are linearly independent by construction.)

Section 5.5 : DegreeEdit

Section 5.6 : The Hilbert FunctionEdit

Chapter 6 : SmoothnessEdit

Chapter 7 : Birational GeometryEdit

Chapter 8 : Maps to Projective SpaceEdit

Section 8.1 : Embedding a Smooth Curve in Three-SpaceEdit

  • Apparently the intention is that $ \phi_x(\lambda) = \varphi_x(\lambda) $ here (both are versions of the lowercase Greek letter phi), which confused me briefly since I assumed one or the other must refer to the parametrization of X.

Section 8.2 : Vector Bundles and Line BundlesEdit

  • Definition of vector bundle: the definition says "the diagram commutes," without indicating a diagram. What's going on would be obvious except for the badly-placed page break -- the diagram it's talking about is at the top of the next page.
  • They start talking about sections here before they're defined in 8.3.
  • I guess the "atlas" functions of a vector bundle are called "local trivializations" because $ U_i \times \mathbb{C}^n $ is trivially a vector bundle, which would make the functions $ (\varphi_i)|_{\pi^{-1}(U)} $ "local isomorphisms to trivial bundles."

Section 8.3 : The Sections of a Vector BundleEdit

Section 8.4 : Examples of Vector BundlesEdit

  • The tautological line bundle: We apparently need to assume that X has positive dimension -- if our projective variety is a single point then we do seem to have a nontrivial section of the tautological bundle.
  • The tautological line bundle: This example is defined for a projective variety X (rather than for a quasi-projective variety). This resolves the apparent ambiguity between "tautological line bundles have no nonzero global sections" and the fact that the sheaf of sections isn't just the zero sheaf. The hyperplane bundle on the next page, on the other hand, is defined for a quasi-projective variety X.

Section 8.5 : Line Bundles and Rational MapsEdit

Section 8.6 : Very Ample Line BundlesEdit

Appendix : Sheaves and Abstract Algebraic VarietiesEdit