Program:

• 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.