Prof. Alexander F. Ritter, Associate Professor, University of Oxford.

♦ Lecture notes, 1 page per side (version 29, Nov 2018)

♦ Lecture notes, 2 pages per side (version 29, Nov 2018)

♦ Exercise Sheets: sheet 0 -- sheet 1 -- sheet 2 -- sheet 3 -- sheet 4

Affine algebraic varieties, the Zariski topology, morphisms of affine varieties. Irreducible varieties.

Projective space. Projective varieties, affine cones over projective varieties. The Zariski topology on projective varieties. The projective closure of an affine variety. Morphisms of projective varieties. Projective equivalence.

Classical embeddings: Veronese embedding, Segre embedding, Plücker embedding. Grassmannians. Flag varieties.

Products. Fibre products. Categorical products. Pullbacks and pushouts.

Coordinate rings. Hilbert's Nullstellensatz. Correspondence between affine varieties and finitely generated reduced k-algebras. Graded rings and homogeneous ideals. Homogeneous coordinate rings.

Introduction to schemes. Spec and the maximal spectrum Specm. Proj.

Categorical quotients of affine varieties by certain group actions.

Primary decomposition of ideals.

Dimension and codimension. Noether normalisation. Every irreducible variety is birational to a hypersurface.

Discrete invariants of projective varieties: degree, dimension, Hilbert polynomial. 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. Derivative map 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.

___________________________

Prof. Alexander F. Ritter. Contact me.

Associate Professor in Geometry, Mathematical Institute, Oxford.

The Roger Penrose Fellow and Tutor, Wadham College, Oxford.