Algebraic Geometry Reading Group (2012-2013)

Session 1: Hilbert basis theorem

Goal: If $R$ is a Noetherian ring, then $R[X_1,\dots,X_n]$ is a Noetherian ring.
Content: Chapter 1 of [Ful], see sections 1.1, 1.2, 1.3, 1.4; Gröbner basis; Buchberger's algorithm.
Speaker: (13/11/12: done) Matthew

Session 2: Hilbert's Nullstellensatz

Goal: Let $I$ be an ideal in $k[X_1,\dots,X_n]$ ($k$ algebraically closed). Then $\mathbb{I}(\mathbb{V}(I))=\mathrm{Rad}(I)$.
Content: Chapter 1 of [Ful], see sections 1.5, 1.6, 1.7.
Speaker: (20/11/12: done) Bryce

Session 3: Fields Extension

Goal: If a field $L$ is ring-finite over a subfield $K$, then $L$ is module-finite (and hence algebraic) over $K$.
Content: Chapter 1 of [Ful], see sections 1.8, 1.9, 1.10; Fundamental theorem of Galois theory.
Speaker: (27/11/12: done) Rémy

Session 4: Affine Varieties

Session 5: Rational Functions and Local Rings

Session 6: Discrete Valuation Rings

Session 2bis: Effective Hilbert's Nullstellensatz

Goal: How the Hilbert's Nullstellensatz can be used in computer science?
Content: Effective Nullstellensatz (slides used under the authorization of Carlos D'Andrea)
Speaker: (15/01/13: done) Igor

Session 7: Quotient Modules and Exact Sequences

Goal: (Dimension theorem) Let $0 \to V_1 \to \dots \to V_n \to 0$ be an exact sequence of finite-dimensional vector spaces. Then $\sum_i (-1)^i \mathrm{dim}(V_i) = 0$.
Content: Chapter 2 of [Ful], see sections 2.7, 2.8, 2.9, 2.10, 2.11; the isomorphism theorem; the splitting lemma.
Speaker: (22/01/13: done) Rémy

Session 8: Multiplicities and Local Rings

Session 9: Intersection Numbers

Session 10: Projective Varieties

Session 11: Affine and Projective Varieties

Goal: Understanding the relationships between affine and projective algebraic geometry.
Content: Chapter 4 of [Ful], see sections 4.3, 4.4.
Speaker:(26/02/13: done) Gaëtan

Session 12: Projective Plane Curves

Session 13: Linear Systems of Curves

Session 14: Bézout's theorem

Goal: Let $F$ and $G$ be projective plane curves of degree $m$ and $n$, respectively. Assume $F$ and $G$ have no common component. Then $\sum_P I(P,F \cap G) = mn$.
Content: Chapter 5 of [Ful], see sections 5.3.
Speaker: (26/03/13: done) Matthew

Session 15: Max Noether's Fundamental Theorem

Goal: Let $F$, $G$, $H$ be projective plane curves. Assume $F$ and $G$ have no common components. Then there is an equation $H = AF + BG$ if and only if Noether's conditions are satisfied at every point in $F \cap G$.
Content: Chapter 5 of [Ful], see sections 5.4, 5.5; Elliptic curves; Cryptography.
Speaker: (date: tbd) Gaëtan

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