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{\Large Introduction to Algebraic Geometry} \\
Math 621 \\
Hurley 168, MWF 11:45 - 12:35 (subject to change) \\
K. Chandler
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Classical algebraic geometry is the study of sets (called varieties)
of solutions
of systems of polynomial equations.
The nature of algebraic geometric inquiry is to relate the geometric
features of a variety to the algebraic properties of the polynomials
describing it.
In modern algebraic geometry, the notion of variety is generalized
to that of a scheme, whereby an arbitrary commutative ring may be
associated with a ``geometric'' object.
In addition to its intrinsic appeal as a subject of study,
algebraic geometry is closely related to commutative algebra, complex analysis,
and number theory, and has applications to PDE, mathematical physics,
topology, geometric optimization, and engineering.
We shall introduce basic notions
of classical algebraic geometry. Our main objects of
consideration will be varieties in affine or projective space.
\begin{itemize}
\item Basic objects: a variety and its coordinate ring
\item Examples: rational normal curves, Veronese and Segre varieties,
Grassmannians, flag varieties
\item attributes of varieties, such as dimension, degree,
smoothness, irreducibility
\item classical constructions and techniques
\item birational maps, blow-ups, resolution of singularities
\item Bertini theorems
\item divisors, linear systems, intersection theory, Riemann-Roch theorems
\end{itemize}
\subsection*{Prerequisite:}
A graduate course in algebra and/or the
candidacy syllabus for algebra is strongly recommended as background.
Specifically, it is essential to have some familiarity with
the following notions: rings, homomorphisms, ideals;
factorization, PID's, UFD's; prime ideals, local rings, localization;
polynomial rings, formal power series rings, Hilbert's basis theorem;
modules, exact sequences, tensor products;
fields, algebraic extensions, transcendence bases;
Noetherian rings and modules.
Also, it is useful to have acquaintance with integral ring extensions,
Noether's normalization lemma, and Hilbert's Nullstellensatz.
A reference for these topics is: Hungerford, {\it Algebra},
Chapters III, IV, V, VI, and VIII.
\subsection*{Main reference:} Shafarevich, {\it Basic algebraic geometry~1}
\subsection*{Additional references:}
Eisenbud, {\it Commutative algebra with a view toward algebraic geometry}\\
Harris, {\it Algebraic geometry} \\
Hartshorne, {\it Algebraic geometry} \\
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\section*{ Spring 2002}
We introduce the language and techniques of modern algebraic geometry.
Topics will include:
\begin{itemize}
\item abstract varieties, vector bundles, sheaves and sheaf cohomology
\item spectra of rings, schemes and functors of points
\item a glimpse of arithmetic algebraic geometry
\item Hilbert polynomials and syzygies
\item families of varieties, parameter spaces: Hilbert schemes
and moduli spaces
\item advanced topics in curve theory
\end{itemize}
\subsection*{Main reference:} Shafarevich, {\it Basic algebraic geometry~2}
\subsection*{Additional references:}
Eisenbud and Harris, {\it Schemes: The language of modern algebraic geometry} \\
Hartshorne, {\it Algebraic geometry} \\
Mumford, {\it The red book of varieties and schemes}
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