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.
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.
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)