# A.Gorodentsev

## Algebraic geometry: a start-up course

This is not an IUM course; what follows is lecture notes (in
English) of the course given by our professor Alexey Gorodentsev
at the University of Warwick (UK) in the Spring semester of 1999.

## Lecture notes in English

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## Syllabus

- Polynomial algebra, affine space, and projective space
associated with a given vector space.
- Projective spaces: coordinates, affine covering, and subspaces.
- Projective spaces: projections and line projective isomorphisms.
- Projective quadrics: the basics.
- Projective quadrics: complex plane conics.
- Projective quadrics: some drawings.
- Projective quadrics: linear subspaces on a non singular quadric, Segre.
- Tensor guide (instead of lecture 8).
- Grassmannian polynomials: computation examples.
- G(2,4) and 3D line geometry.
- Grassmannians in general.
- Contractions and polarizations.
- Linear span of a tensor. Plücker relations.
- Partial derivatives and Veronese varieties.
- Geometry of Veronese curve.
- Projective hypersurfaces.
- Plane curves: intersections.
- Plane curves: singular points and tangents.
- Plane curves: Plücker formulas and duality.
- Plane curves: Chasles - Cayley - Brill formula.
- Commutative algebra draught.
- Algebraic - geometric dictionary.
- Algebraic manifolds.
- Some morphisms.