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.

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.

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.

Properties of morphisms of schemes

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