I plan to cover lots of basic commutative algebra, at the level of Atiyah-Macdonald or Miles Reid's UCA, but my emphasis will be on context and historical motivation. The context is the theory of algebraic curves. The historical motivation is Dedekind and Weber's purely algebraic treatment of Riemann surfaces in the language of algebraic number theory. I will cover higher dimensional geometry only as it relates to these topics.
Lecture: 12:30-1:45 TuTh on Zoom
Office Hours: After the lecture
Here are the typed lecture notes. The handwritten notes are below
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| Assignment |
Date |
Handwritten Notes |
Introduction
Homework 1
Solutions
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Mon, Sept 14
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Examples of curves
Some famous theorems
More famous theorems
Equivalence of curves
Homogeneous coordinates
Projective equivalence
(Maple worksheet)
Fundamental theorem of projective geometry
Diagonalization of quadratic Forms
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PIDs
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Fields and domains
Maximal and prime ideals
Principal Ideal Domains
PID implies UFD
Applications to Z and F[x]
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Tangent Spaces
Homework 2
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Mon, Oct 12
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Homogeneous polynomials
Formal derivatives and the chain rule
Taylor expansion
Tangent spaces to hypersurfaces
Projective space in general
Projective subspaces and duality
Projective Descartes theorem, line intersect hypersurface
Affine vs projective tangent spaces
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Nullstellensaetze
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Gauss' Lemma
Study's Lemma for curves
Study's Lemma for hypersurfaces
Ideal of a point, Zariski tangent space
Sylvester's Resultant
Nullstellensatz, Classical Form
Noetherian Rings, Hilbert Basis Theorem
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Zariski Topology
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Galois Connections
Zariski Topology
Irreducible Varieties
Projective Varieties
Projective Zariski Topology
Projective Completion
Twisted Cubic Curve I
Twisted Cubic Curve II
Preview of Next Semester
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