Магистратура 1 курс. Algebraic Geometry. Start Up Course

Лектор: профессор А.Л.Городенцев

 

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Prerequisits

Basic linear and multilinear algebra (tensor products, polylinear maps) and basic commutative algebra (polynomial rings and their ideals) are necessary. Some experience in geometry and topology (linear affine geometry, topological manifolds) is desirable but not so essential.

Contents of the coutse

Projective Geometry

Projective spaces.

Projective conics and PGL₂

Geometry of projective quadrics. Spaces of quadrics

Grassmannians.

Examples of projective maps: Pluecker, Segre, Veronese.

Plane projective algebraic curves: point multiplicities, intersection numbers, Bezout's theorem, singularities, duality, Pluecker relations, rational curves, Veronese curves, cubic curves.

Affine Algebraic Geometry

Integer ring extensions, polynomial ideals, and Hilbert's theorems.

Algebraic varieties, Zarisky topology, schemes, geometry of ring homomorphisms.

Irreducible varieties. Products. Dimension.

Algebraic Manifolds

Sheaf of regular functions. Separable and proper manifolds. Affine and projective manifolds.

Curves on surfaces. The 27 lines on a smooth cubic surface.

Linear systems and invertible sheaves, the Picard group, line bundles on affine and projective spaces.

Vector bundles and their sheaves of sections. Vector bundles on the projective line.

Tangent, cotangent, normal and conormal bundles. Singularities and tangent cone. Blow up.

Lecture notes and textbooks

• Lecture 1. Projective spaces. (gzipped PS or PDF made from PS).
• Lecture 2. Projective quadrics. (gzipped PS or PDF made from PS).

 

Besides the lecture notes presented above the following text books are recommended:

1. C.H.Clemens, A scrapbook of complex curve theory, Plenum Press, 1980
2. J.Harris, Algebraic geometry. A first Course, Springer, 1998
3. M.Reid, Undergraduate algebraic geometry, CUP, 1988
4. I.R.Shafarevich, Basic algebraic geometry, Vol 1, Springer 1994

Home tasks

• Task 1 (February 7). Projective spaces (gzipped PS or PDF made from PS).
• Task 2 (February 14). Linear automorphisms of projective line, conics, and polarities (gzipped PS or PDF made from PS).
• Task 3 (February 21). Projective quadrics (gzipped PS or PDF made from PS).
• Task 4 (February 28). Pluecker–Segre–Veronese interaction (gzipped PS or PDF made from PS).
• Task 5 (March 21). Commutative algebra draught (gzipped PS or PDF).
• Task 6 (April 4). Curves (gzipped PS or PDF).
• Task 7 (April 18). Algebraic manifolds (gzipped PS or PDF).
• Task 8 (April 25). 27 lines (gzipped PS or PDF).

Exam and marks

Tere is a written exam after this course. The final mark is calculated from the ammounts of solved exam problems and solved problems from the home tasks. If you have solved T% of home task problems and E% of exam problems, then to take the mark "A" it is enough to collect T+E=120 or more. Here is a real variant of the final exam problems: gzipped PS.

There is also an obligatory midterm test (aprox. in the middle of March). It makes no contribution to the final mark and should be considered as a rehearsal of the final written exam (it allows to make necessary corrections if something goes not entirely as it was planned). Here is a real variant of the midterm test problems: gzipped PS.