**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.