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HW#5 (Due 11/4 @ 5pm):
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HW#6 (Due 11/11 @ 5pm):
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HW#7 (Due 11/18 @ 5pm):
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HW#8 (Due 11/25 @ 5pm):
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In problem (1), "not
intersecting" should be "intersecting".
HW#9 (Due 12/3 @ 5pm):
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The prerequisites are a good course in
Algebra and enthusiasm. Thanks to the large enrollment, we will
have a T.A. (Matthew Gealy), and he will conduct a problem session every Friday between 2 and 3
PM in 153 Sloan. Everyone is urged to attend. If this time is bad, go to the first (Friday) meeting and work out a suitable time.
This introductory course is aimed at advanced undergraduates,
beginning graduate students, physicists and other non-experts who
may want to gain a basic understanding of the subject. We will use
Serre's notion of an algebraic variety, which is more modern and
more understandable than that of Weil, but we will not appeal to
the later, very powerful scheme-theoretic language of that
visionary called Grothendieck; hopefully some of that will be
introduced in the second quarter. We will work over algebraically
closed fields k and use Zariski topology, which is coarser than
the standard topology when k = C, but which suffices for our
purposes and allows for a uniform presentation. One could simply assume
throughout that k = C; there is a parallel theory of "analytic
varieties,'' which will be mentioned only in passing.
The case k = \overline Fp is also of
interest, equipped with the Frobenius action, particularly for those
interested in Algebra or Combinatorics.
The following is a rough outline of what we hope to cover:
- Noetherian spaces, connectedness, irreducibility, constructibility
- Affine and Projective spaces, transformations, Segre embedding,
Grassmannians
- Noether normalization, Cohen-Seidenberg's going up theorem
- Algebraic sets, Hilbert's Nullstellensatz
- Zariski topology, examples
- Algebraic Curves, rational functions
- Presheaves, sheafification, regular functions
- Ringed spaces, varieties, maps, products
- Points: closed, geometric, generic, rational
- Dimensionality, singular points, normal and smooth varieties
- Complete varieties, projectivity
- Birational maps, Zariski's main theorem
- Line bundles, divisors, ampleness
Text: The recommended text for the course is The Red Book of
Varieties and Schemes by David Mumford. The lectures will only loosely follow it,
however, and mainly just the first chapter. The students are urged to follow
along with the text.
Grades:
There will be no midterm or final examination. There will be
weekly homework exercises, and the entire grade will depend on
solving these problems. The students are allowed to collaborate
and also discuss with the T.A. about any difficult or unclear
point, but the write-up of solutions must be done individually.
Contact Info:
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