### 2020–21–A

## Prof. Ilya Tyomkin

#### Time and Place:

- יום ג 11:00 - 09:00
- יום ה 14:00 - 12:00

## Course topics

- Affine and projective spaces, affine and projective maps, Segre and Veronese embeddings, Desargues’s Theorem, Pappus’s Theorem, cross-ratio, projective duality
- Plane curves: rational curves, linear systems of curves, conics and the Butterfly Theorem, Pascal’s Theorem, Chasles’s Theorem, the group structure on a planar cubic, Bezout’s Theorem
- Affine algebraic varieties: Hilbert’s Basis Theorem, Zariski topology, irreducible components, Hilbert’s Nullstellensatz, the correspondence between the ideals and the algebraic sets, morphisms and rational maps between affine algebraic varieties
- Projective varieties: graded rings and homogeneous ideals, the projective correspondence, morphisms, blow-ups, birational equivalence and rational varieties, Grassmannians
- The basics of dimension theory
- The basics of smoothness
- Cubic surfaces and 27 lines. If time permits, other topics will be discussed such as abstract algebraic varieties, Chevaley’s Theorem, Riemann-Roch Theorem and its applications.