PMATH 764: Algebraic curves
PMATH 764: Algebraic curves
Lectures: MWF 9:30-10:20 (MC 4041).
Office hours: T 10:00--12:00 and Th 15:00--17:00.
Course information:
Outline.
Course web page:
https//:learn.uwaterloo.ca.
Course description:
An introduction to the geometry of algebraic curves with applications to elliptic curves and computational algebraic geometry. Plane curves, affine varieties, the group law
on the cubic, and applications.
Objectives:
This course is in fact an introduction to algebraic geometry through the theory of algebraic curves. The main
objective of the course is to present some of the basic concepts and
techniques of algebraic geometry, with an emphasis on how these specialise
to algebraic curves.
Outline of topics:
- General algebraic geometry:
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Affine and projective varieties.
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Coordinate rings, fields of functions, and local rings.
-
Polynomial and rational maps.
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Dimension and smoothness.
-
Algebraic curves:
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Affine and projective plane curves, and their local properties.
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Intersection numbers and Bézout's Theorem.
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Divisor class groups.
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Elliptic curves and the group law on the cubic.
Prerequisites:
PMATH 345 -- Polynomials, rings and finite fields or
PMATH 347 -- Groups and Rings
(or the equivalent).
References:
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Algebraic curves: an introduction to algebraic geometry, by W.
Fulton;
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Algebraic geometry, by R. Hartshorne;
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Algebraic curves and Riemann surfaces, by R. Miranda;
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Algebraic geometry: a first course, by J. Harris;
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The red book of varieties and schemes, by D. Mumford;
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Basic algebraic geometry, by I. R. Shafarevich.