C3.4 Algebraic Geometry (2022-23)
Main content blocks
- Lecturer: Profile: Damian Rossler
B3.3 Algebraic Curves is useful but not essential. Projective spaces and homogeneous coordinates will be defined in C3.4, but a working knowledge of them would be useful. There is some overlap of topics, as B3.3 studies the algebraic geometry of one-dimensional varieties.
Courses closely related to C3.4 include C2.2 Homological Algebra, C2.7 Category Theory, C3.7 Elliptic Curves, C2.6 Introduction to Schemes; and partly related to: C3.1 Algebraic Topology, C3.3 Differentiable Manifolds, C3.5 Lie Groups.
Projective space. Projective varieties, affine cones over projective varieties. The Zariski topology on projective varieties. The projective closure of affine variety. Morphisms of projective varieties. Projective equivalence.
Veronese morphism: definition, examples. Veronese morphisms are isomorphisms onto their image; statement, and proof in simple cases. Subvarieties of Veronese varieties. Segre maps and products of varieties.
Coordinate rings. The geometric form of Hilbert's Nullstellensatz. Correspondence between affine varieties (and morphisms between them) and finitely generated reduced K-algebras (and morphisms between them). Graded rings and homogeneous ideals. Homogeneous coordinate rings.
Discrete invariants of projective varieties: degree, dimension, Hilbert function. Statement of theorem defining Hilbert polynomial.
Quasi-projective varieties, and morphisms between them. The Zariski topology has a basis of affine open subsets. Rings of regular functions on open subsets and points of quasi-projective varieties. The ring of regular functions on an affine variety is the coordinate ring. Localisation and relationship with rings of regular functions.
Tangent space and smooth points. The singular locus is a closed subvariety. Algebraic re-formulation of the tangent space. Differentiable maps between tangent spaces.
Function fields of irreducible quasi-projective varieties. Rational maps between irreducible varieties, and composition of rational maps. Birational equivalence. Correspondence between dominant rational maps and homomorphisms of function fields. Blow-ups: of affine space at a point, of subvarieties of affine space, and of general quasi-projective varieties along general subvarieties. Statement of Hironaka's Desingularisation Theorem. Every irreducible variety is birational to a hypersurface. Re-formulation of dimension. Smooth points are a dense open subset.
Section outline
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There will be two consultation sessions run by the lecturer (Damian Rössler) in TT. The schedule is
- W2 Mo 4.30-6.30pm in C1
In the first hour, we will go through Q1 of the 2021 paper, which can be done with the material covered in MT22 (not all the exams of the previous years can). The second hour will be reserved for questions.
- W3 Mo 4.30-6.30pm in C4.
In the first hour, we will go through Q2 of the 2021 paper. The second hour will be reserved for questions.
- W4 Fri 4-6pm in C1.
We will go through various questions from old papers.
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Please upload your solution to Sheet 1 here.
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Please upload your solution to Sheet 2 here.
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Please upload your solution to Sheet 3 here.
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Please upload your solutions for Sheet 1 here.
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Please upload your solutions for Sheet 2 here.
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Please upload your solutions for Sheet 3 here.
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Please upload your solutions for Sheet 1 here
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Please upload your solutions for Sheet 2 here
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Please upload your solutions for Sheet 3 here.
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