# Algebraic Geometry Notes

This page collects all of the expository notes I have written on algebraic geometry. These include self-contained summaries of particular topics, and looser lecture notes offering motivation for concepts or glimpses of more advanced topics, as well as cheat sheets of results and definitions. Many of the notes are somewhat rough, and no guarantees are made as to accuracy. Also, notes from my courses (Fall 2006, Spring 2007, Fall 2009, Winter 2010) may reference problem sets or such.

## Topic Overviews

These notes give self-contained introductions to various topics in foundational algebraic geometry, generally attempting to give arguments or references for all assertions. The earlier notes deal with varieties from a semi-classical perspective, while the later ones deal with schemes.

Recovering geometry from categories
Explains how the underlying set and Zariski topology of a variety can be recovered solely from the category.

Points and morphisms
Discusses the classical and scheme-theoretic perspective on points and morphisms, and the relationship between them, in both the affine and projective cases.

Varieties via atlases
Gives a definition of abstract varieties over algebraically closed fields using atlases.

Projective varieties
A development of projective varieties as examples of abstract varieties, including a study of regular functions on projective varieties and morphisms to projective varieties. Also, a brief discussion of completeness.

Nonsingular curves
A basic presentation of nonsingular curves over an algebraically closed field, including quasiprojectivity and the nonsingular projective compactification. To this end, discusses normality and normalization.

Divisors on nonsingular curves
We introduce the concept of divisors on nonsingular curves, and study their properties and their relationship to morphisms to projective space.

Chevalley's theorem and complete varieties
Gives a proof of Chevalley's theorem on the constructibility of the image of a morphism of varieties. Discusses completness, and gives the variety version of the valuative criterion, in terms of extending morphisms from nonsingular curves.

Differential forms
A presentation of differential forms on abstract varieties over algebraically closed fields, with applications to the ramification of a morphism of nonsingular curves.

The Riemann-Roch and Riemann-Hurwitz theorems
We state the Riemann-Roch theorem without proof, and give a brief sketch of some applications, including a proof of the Riemann-Hurwitz theorem.

Secant varieties and curves in projective space
A brief discussion of secant varieties, applied to projective of curves from higher-dimensional projective space to three-space and to the plane.

Power series and nonsingular points
A discussion of the complete local ring of a variety, and its basic consequences for singularity and nonsingularity, including the fact that a nonsingular point is a local complete intersection.

Properties of fibers and applications
Studies irreducibility and reducedness of schemes over fields, and the behavior under extension of base field. Examines how these ideas relate to fibers of morphisms, and finally applies the results to prove that any two (closed) points of a variety can be connected by a (not necessarily irreducible) connected curve.

Complex varieties and the analytic topology
Introduces the analytic topology on complex (pre)varieties, and proves basic properties, including the equivalence of Hausdorff and the variety condition, of compact and complete, and of connectedness in the two topologies. Also studies the significance of nonsingularity.

Yoneda's lemma and representable functors
Introduces representable functors, motivated by the desire to formalize the notion of a moduli space. Introduces variants to obtain more general universal properties.

Zariski sheaves and the fiber product
Introduces Zariski sheaves and the basic criterion for representability by finding a cover by representable open subfunctors. Applies this to proving the existence of fiber products. Briefly discusses the behavior of fiber products of schemes.

Properties of properties of morphisms
A note intended to help the reader organize common behaviors for properties of morphisms.

Valuative criteria
An essentially self-contained presentation of the valuative criteria for separatedness, universal closedness, and properness, which attempts to present minimal hypotheses, and organize the ideas behind the proofs as clearly as possible.

Dimension theory
Discusses the pathologies of dimension theory even for Noetherian schemes, and ways in which one can phrase dimension-theoretic arguments so that they go through nonetheless.

Closed immersions
An almost completely general criterion for closed immersions in terms of injectivity of points and tangent vectors, with conclusions drawn for varieties and curves, and consequent applications of Riemann-Roch.

Associated points
A development of associated points, imbedded points, and multiplicities, with an eye towards Hilbert polynomials.

Hilbert polynomials
Hilbert polynomials, degrees, and a generalized Bezout's theorem.

Cohomology and base change
Collects general results on the behavior of cohomology sheaves under base change. Primarily intended to help track down references for statements more general than those found in Hartshorne.

Schlessinger's criterion
Lecture notes from the MSRI workshop on deformation theory, primarily treating Schlessinger's criterion but also including a more informal discussion of some additional topics.

## Lecture notes

These are notes based on lectures I've given in classes or elsewhere on more advanced topics in algebraic geometry. On the whole these are less comprehensive, settling for giving a conceptual sketch of the topic in question rather than providing careful definitions and statements.

Algebraic geometry through the lens of elliptic curve theory
An attempt to survey as many concepts from algebraic geometry as possible via relatively elementary elliptic curve theory.

Why schemes?
An attempt to motivate scheme theory.

Fixing the Zariski topology
Gives a detailed motivation from the perspectives of point-set topology and the analytic topology for the definitions of separatedness and properness.

An introduction to deformation theory
Gives a brief overview of deformation theory as a tool of moduli theory, intended as a motivation for scheme theory and an invitation to Cech cohomology.

Intersection theory
An introduction to intersection theory, motivated mainly by enumerative geometry.

The minimal model program
A sketch of the minimal model program, focusing mainly on the surface and threefold cases.

The Weil conjectures and etale cohomology
The well-worn tale of the Weil conjectures and the consequent development of the theory of etale cohomology.

Quot and Hilbert schemes
A description of the Quot and Hilbert schemes, together with a sketch of proof of representability, and application to Hom schemes.

Deformations of subschemes and morphisms
A local description of Hilbert and Hom schemes via deformation theory.

Mori's bend-and-break argument
An overview of Mori's bend-and-break argument for the existence of rational curves on Fano varieties.

Two degeneration techniques for maps of curves
Compares and contrasts the theories of admissible covers and limit linear series.
Published in: Snowbird Lectures in Algebraic Geometry, 137-143, Contemporary Mathematics 388, AMS, 2005.

Connections, curvature, and p-curvature
Examines algebraic connections from both the classical and Grothendieckian point of view, including Mochizuki's definition of p-curvature in the latter setting.

## Cheat sheets

These are condensed cheat sheets collecting definitions and statements in broad categories, with citations for the proofs. For now, they are based primarily on Hartshorne, although with some substantial supplementary material from elsewhere.

Cohomology of sheaves (including first applications of Riemann-Roch to curve theory)