Projective Geometry (as Algebraic Geometry 2)
Independent University of Moscow, Spring 2014
Synthetic and algebro-geometric views on the projective plane, incidence relation, projective duality.
Projective space, Grassmannian, Veronese embedding, Segre embedding, Veronese variety.
Every projetive variety is isomorphic to an intersection of quadrics.
Morphisms and rational maps of projetive varieties. Veronese embeddings and projections.
Graded rings and their projective spectra.
Maps to projective spaces, line bundles, divisors and their classes, linear systems.
Dimension, codimension, degree. Theorem of Bertini and del Pezzo.
Determinantal varieties. Secant varieties. Projective duality.
Cohomology of line bundles and syzigies. Normality.
Canonical bundle. Hyper-elliptic and canonical curves.
Projective models of curves of low genera.
Projective models of some rational surfaces.
Rationality constructions.