PMATH 764: Algebraic curves
PMATH 764: Algebraic curves
Lectures: MWF 9:30--10:20 (MC 4058).
Office hours: W 13:00--15:00, F 13:00--14:30.
Course information:
Outline.
Course web page:
http//:uwace.uwaterloo.ca.
Course description:
An introduction to the geometry of algebraic curves with applications to
elliptic curves and computational algebraic geometry.
Objectives:
This course is in fact an introduction to algebraic geometry. 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. A secondary objective is to illustrate some of the
links between algebra and differential geometry in the
study of the geometric properties of algebraic curves.
Syllabus:
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Affine varieties;
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Plane curves and their local properties (e.g., multiple points,
tangent lines, intersection numbers);
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Projective space and projective varieties;
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Projective plane curves and their intersections (Bézout's
Theorem);
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Divisors and Jacobians of curves;
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Rational and elliptic curves;
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The group law on the cubic.
Prerequisites:
PMATH 345 -- Polynomials, rings and finite fields.
References:
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Algebraic curves: an introduction to algebraic geometry, by W.
Fulton;
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Basic algebraic geometry, by I. R. Shafarevich;
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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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Introduction to algebraic curves, by P. Griffiths.
Assignments: