This is the current working version of the course. It is updated during the term-time. You can also look at the old stable version of this course (unchanged after 2009). Besides my own
lecture notes, which will appear as far as will be ready, I recomend the following textbooks:
- A.L.Gorodentsev, Algebraic Geometry. A Start Up Cource, MCCME. You can download release Feb. 16, 2004, size: 825 kB or release Aug. 28, 2006, size: 896 kB.
- A.L.Gorodentsev, Algebra I. Textbook for Students of Mathematics. Springer, Ch. 11.
- A.L.Gorodentsev, Algebra II. Textbook for Students of Mathematics. Springer, Ch. 1, 2, 10, 11, 12.
- J.Harris, Algebraic Geometry. A First Course, Springer.
- M.Reid, Undergraduate algebraic geometry, CUP.
All currently available lectures via one file (PDF, 1.3 Mb, updated 2017.11.24).
- Lecture 1. Projective spaces. (Updated 2017.10.11.) Affine and projective algebraic varieties. Spaces of hypersurfaces. Veronese curves. Projections and linear projective isomorphisms. The cross-ratio and harmonicity.
- Lecture 2. Projective quadrics. (Updated 2017.09.21.) Tangent lines and singular points. The polar map and projective duality. Spaces of quadrics. Conics and qudratic surfaces in more detailes. Linear subspaces laying on a quadric.
- Lecture 3. Working examples: lines and conics on the plane. (Updated 2017.09.21.) Homograhies between lines, pencils of lines, and conics. Cross-axes. Inscribed and circumscribed triangles and hexagons. Internal geometry of a smooth conic. Pencils of conics.
- Lecture 4. Tensor guide. (Updated 2017.10.07.) Tensor products and tensor powers. Pairings and contractions. The linear support of a tensor. Symmetric and grassmannian algebras. Symmetric and skew symmetric tensors. The polarization of polynomials. Segre's, Verenose's, and Grassmannian varieties. Decomposable grassmannian polynomials, the Plücker embedding.
- Lecture 5. Grassmannians in more details. (Updated 2017.11.24.) The grassmanian Gr(2,4), Plücker quadric, and lines in P3. The lagrangian grassmanian LGr(2,4) and lines on a smooth quadric in P4. The Grassmanians Gr(k,n): the Plücker embeding; homogeneous, Plücker's, and local affine coordinates; the cell decomposition and Schubert cycles.
- Lecture 6. Commutative algebra draught. (Updated 2017.11.24.) Noetherian rings, integral and algebraic elements, normal rings, finitely generated algebras over a field, transcendence generators. Systems of polynomial equations, the Hilbert Nullstellensatz, resultants.
- Lecture 7. Affine algebraic geometry. (Updated 2017.11.24.) Affine Algebraic–Geometric Dictionary: maximal spectrum and pullback morphisms. Zariski topology. Rational functions and structure sheaf. Geometric properties of algebra homomorphisms.
- Lecture 8. Algebraic manifolds. (Updated 2017.11.24.) Definition and examples. Separated manifolds. Regular and rational maps. Projective manifolds, closeness of projective maps. Finite projections, Noether's normalization.
- Lecture 9. Dimension. (Updated 2017.11.24.) Combinatorial and transcendental dimenson. Dimensions of submsnifolds. Dimensions of fibers of regular maps. Dimensional criterion of irreducibility. Dimensions of projective varieties. Applications: the resultannt of a system of n homogeneous polynomial equations in n unknowns, lines laying on surfaces, 27 lines on a smooth cubic surface.
The problems are either obligatory or honorary (optional). The obligatory problems are those not marked with stars from the tasks numbered by integers. The problems marked with stars and all problems from the tasks numbered by non-integers (e.g., 2½) are honorary. The maximal final mark «A» (or 10 for the HSE students) can be achieved without solving the honorary problems.
Let H and E be the total amounts of problems you have solved among the Home Tasks and during the Written Exam, both computed as percentage [total number of solved problems]:[total number of obligatory problems] (this may be >100 if you solved honorary problems). To get the marks «A», «B», «C» it is enough to have H+E equal to at least 140, 90, 40 respectively. For the HSE students, the final mark equals min(140,H+E)/14.