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Algebraic Geometry Start Up Course. Spring term 2011.

by A.L.Gorodentsev

Prerequisits   Contents   Lecture notes   Home tasks   Exam and marks  


Basic linear and multilinear algebra (vector spaces, tensor products, multilinear 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.


Projective Geometry
Projective spaces.
Projective conics and PGL2
Geometry of projective quadrics. Spaces of quadrics
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 recommended textbooks

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

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.

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