Syllabus | |
Week |
-----------------------------------------Topics Covered----------------------------------------- |
1 |
Topological Spaces: Open, closed,
irreducible, dimension, examples |
2 |
Affine Varieties: Polynomial ring $k[x_1,\dots,x_n]$, ideals, zero sets of ideals, ideal of subsets of $\mathbb{A}^n$, Zariski topology on $\mathbb{A}^n$ |
3 |
Functions on affine varieties:
coordinate ring, local ring, regular and rational
functions, morphism between varieties and morphisms
between their coordinate rings |
4 |
Smooth vs Singular: analytic
definition of smoothness and algebraic counterpart,
regular local rings |
5 |
Projective varieties: graded ring
$k[x_0,\dots,x_n]$, homogeneous ideals, projective
algebraic sets, Zariski topology on $\mathbb{P}^n$,
projective coordinate ring. |
6 |
Quasi-projective varieties:embedding
affine varieties into projective space, projective
closure, products of varieties, intersection in
projective space |
7 |
Blow-up map:construction of
blow-up space, local charts, resolution of singularities
of a singular curve |
8 |
Arf Rings:multiplicity sequence,
blow-up, du Val characters, Arf rings and Arf closure,
arithmetic of Arf rings |
9 |
Grassmannians: affine affine
charts, Schuberts cells, homology, Schubert calculus,
Plücker embedding, incidence variety and its dimension |
10 |
Presheaves and sheaves: morphisms,
kernel, cokernel, image, sheafification, direct and
inverse images |
11 |
Schemes: Spectrum of $A$, ringed
spaces and morphisms |
12 |
Affine schemes: patching
spectrums, affine line, affine plane, generic point,
affine line with a double point |
13 |
Projective schemes: $Proj$ of a
grades ring, schemes over a base scheme |
14 |
First properties of schemes:topological
properties, open and closed schemes, sheaves of modules,
Weil and Cartier divisors, invertible sheaves |
15 |
Cech cohomology |
It has been estimated that, at the present state of our knowledge, one could give a 200 semester course on commutative algebra and algebraic geometry without ever repeating himself. So any introduction to this subject must be highly selective.Our selection in 2019 is intended to define the main objects of study in algebraic geometry and their refinements, and give some immediate applications.
The weakness of these notes is what had originally driven me to undertake the bigger project: there is no real theorem in them!(Here the bigger project that Mumford refers to, in 1988, is the two volume book on an introduction to algebraic geometry, the second volume of which has appeared only in 2015 with the collaboration of Tadao Oda.)
Exam |
Due Date |
Solution |
Exam01 |
5 November 2019 |
Solution |
Exam02 |
7 November 2019 |
Solution |
Exam03 |
5 December 2019 |
Solution |
Exam04 |
5 December 2019 |
Solution |
Exam05 |
12 December 2019 |
Solution |
Exam06 |
31 December 2019 |
Solution |
Final Exam |
10 January 2020 |
Solution |