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Algebra and Number Theory (MATH596, Winter 2015):
A second course in
algebraic geometry: curves and surfaces.
Prof. Eyal Goren
Location and time: Monday 12:30 - 14:30 in BURN 1214 and
Friday 9:30 - 11:30 in BURN 1205. Monday 13:30-14:30 is
designated as an exercise session (we will start those after 2
weeks, once we have sufficiently many exercises. Note that we
may "overflow" a bit if needed.)
Office Hours: Monday 11:00-12:00 (BURN 1108), or by
schedule in coming weeks:
Monday, March 16,
12:30 - 15:00 (1:30 - 15:00, exercise session), BURN 1214
Tuesday, March 17,
8:30 - 9:45, BURN 920
Friday, March 20,
9:30 - 11:30, BURN 1205
Week of March 23 -
27, usual schedule
Week of March 30 - April 3, no Friday class (Good Friday)
Week of April 6 - 10, No classes (these were already returned
by our make up classes)
Week of April 13 - 17: usual Monday class AND a class on
Tuesday following Friday schedule.
Syllabus: The goal of the course is to
study in detail two important classes of algebraic
varieties: curves and surfaces. Curves and surfaces are of
course examples of fundamental importance just by the virtue
of being the lowest dimensional examples, except for the
zero-dimensional varieties. However, their importance is
much greater as, for one, moduli spaces of curves are of
fundamental importance to mathematical physics and,
secondly, the Jacobian of a curve is the fundamental example
of an abelian variety (in particular, the curves of genus 1
- the elliptic curves - that are isomorphic to their
jacobian are the bread and butter of number theory). In
addition, surfaces offer a view of many of the deeper
techniques of algebraic geometry, such as intersection
theory, adjunction formula, families of curves, and more.
Our fundamental reference will be Hartshorne/Algebraic
geometry (GTM 52), in particular Chapters IV and V. We
will also use Mumford/Lectures on curves
on an algebraic surface (Princeton University Press)
and Fulton/ Algebraic curves: An
introduction to algebraic geometry. We will first go
rather quickly through the theory of sheaf cohomology. One
cannot really get off the ground without those. After that,
we shall start discussing the material in Hartshorne.
Similar to the first course, although understanding curves
and surfaces sounds like a modest enough goal, there is too
much material there to cover linearly and systematically.
The lectures will thus highlight some key theorems
(Riemann-Roch, Hurwitz, Adjunction, Hodge index, Hilbert
polynomial) and some key examples (rational surfaces,
elliptic curves, curves of genus 2, the cubic surface, K3
surfaces, toric varieties).
It is recommended that students review
the connection between the theory of schemes and classical
algebraic geometry over an algebraically closed field, as
in Mumford's red book, Ch. II.3 and the notion of quasi-coherent
sheaves as in loc. cit., Ch. III.1, or Hartshorne Ch.
II.5. We will actually begin the
course by recalling the sheaf of differentials, going
further than in the first term, and by recalling some of the
material in Hartshorne Ch. I.1-I.5. At that
point, we will take a departure into sheaf cohomology after
which we will study curves and, later, surfaces.
Pre-requisites: MATH 596 Fall 2014, or equivalent
(instructor's permission required).
Method of Evaluation: Exercise sessions
(Participation is mandatory). List of Exercises (typos possible!)
Academic integrity: McGill University values academic
integrity. Therefore, all students must understand the meaning
and consequences of cheating, plagiarism and other academic
offences under the Code of Student Conduct and Disciplinary
Procedures (see www.mcgill.ca/integrity for more information).
Submitting work: In accord with McGill University's Charter of
Rights, students in this course have the right to submit in
English or in French any written work that is to be graded.
Syllabus and Grade Calculation: In the event of
extraordinary circumstances beyond the University's control, the
content and/or evaluation scheme in this course is subject to