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.
Gives a definition of abstract varieties over algebraically closed fields using atlases.
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.
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
We introduce the concept of divisors on nonsingular curves, and study their properties and their relationship to morphisms to projective space.
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.
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
We state the Riemann-Roch theorem without proof, and give a brief sketch of some applications, including a proof of the Riemann-Hurwitz theorem.
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
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
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.
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
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
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
A note intended to help the reader organize common behaviors for properties of morphisms.
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.
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.
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.
A development of associated points, imbedded points, and multiplicities, with an eye towards 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.
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.
An attempt to motivate scheme theory.
Fixing the Zariski
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
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.
An introduction to intersection theory, motivated mainly by enumerative geometry.
The minimal model
A sketch of the minimal model program, focusing mainly on the surface and threefold cases.
The Weil conjectures and
The well-worn tale of the Weil conjectures and the consequent development of the theory of etale cohomology.
Quot and Hilbert
A description of the Quot and Hilbert schemes, together with a sketch of proof of representability, and application to Hom schemes.
subschemes and morphisms
A local description of Hilbert and Hom schemes via deformation theory.
An overview of Mori's bend-and-break argument for the existence of rational curves on Fano varieties.
Two degeneration techniques for maps of
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
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 schemes
Properties of morphisms of schemes
Cohomology of sheaves (including first applications of Riemann-Roch to curve theory)