This is current working wersion filled during the reading of the course. Here is old stable version of this course (unchanged after 2009) and complete old textbooks: release Feb. 16, 2004, size: 825 kB and release Aug. 28, 2006, size: 896 kB.
All currently available lectures via one file (PDF, 878 Kb, renewed 04.10.2014).
- Lecture 1: projective spaces. Affine and projective algebraic varieties. Spaces of hypersurfaces. Veronese curve. Projections and linear projective isomorphisms. Homographies. Cross-ratio and harmonicity.
- Lecture 2: projective quadrics. Tangent lines and singular points. Polar mapping and projective duality. Space of quadrics. Conics and qudratic surfaces in more detailes. Linear subspaces on quadrics. Digression: quadratic forms over arbitrary fields.
- Lecture 3: tensor guide. Tensor products and tensor powers. Duality, contractions, linear support of tensor. Symmetric and grassmannian algebras. Symmetric and skew symmetric tensors, polarization of polynomials. Segre varieties. Decomposable grassmannian polynomials.
- Lecture 4: grassmannians. Grassmanian Gr(2,4), the Plücker quadric and lines in P_{3}. Lagrangian grassmanian LGr(2,4) and lines on a smooth quadric in P_{4}. Grassmanian Gr(k,n): the Plücker embeding, cell decomposition etc.
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All problems marked with stars are honorary. The maximal mark «A» can be achieved without solving them.
- Task 1: acquaintance with projective spaces.
- Task 2: projective quadrics, especially conics.
- Task 3: tensors and Plücker–Segre–Veronese interaction.
- Task 4: projective curves.
- Task 5: affine algebraic geometry.
- Task 6: algebraic maniflds.
- Task 7: around 27 lines and rational mappings.
Let H and E be the total amounts of problems you have solved among the problems given in the Home Tasks and in the Written Exam respectively (both are computed as percentage [total number of solved problems]:[total number of problems without stars] and may be >100 if you solved honorary problems with stars). To get marks «A», «B», «C» it is enough to have H+E > 140, 90, 40. respectively.
Training midterm test (it makes no contribution to the final marks)
Final written exam (December 18, 2014)