240167 Algebraic Geometry (V) (SoSe 2017)

This will be a basic course introducing the tools of modern algebraic geometry.
It will be followed in the WS17/18 by a second course on Algebraic Geometry,
dealing with topics such as Hodge theory, abelian varieties, K3 surfaces,
intersection theory, and moduli spaces. The sequence of these two lectures
is appropriate to be used as "Vertiefung".

Topics to be covered include affine varieties, projective varieties, sheaves, abstract
varieties (over an algebraically closed field), schemes, their properties and the
properties of morphisms between them, blow-ups, rational maps, coherent sheaves,
divisors, linear systems, sheaf cohomology, and differentials.

Along the way, we will establish classical results of algebraic geometry: Chow's
lemma, Bertini theorems, Bezout's theorem, the Riemann-Roch theorem for curves
and for surfaces, etc.

Requirements for participation, required level

Although I will recall the relevant statements and theorems from commutative algebra, I will assume that the students
are familiar with the basic theory of rings and modules, as taught in Professor Voll's course on Commutative Algebra in the WS16/17.

A useful reference is

1. M. Atiyah and I. MacDonald, Introduction to Commutative Algebra, Addison-Wesley (1969)

Bibliography

Most of the course material can be found in the following books.

1. D. Eisenbud and J. Harris, The geometry of schemes.
2. G. Kempf, Algebraic Varieties.
3. R. Hartshorne, Algebraic Geometry.
4. J. Harris, Algebraic Geometry. A first course.
5. D. Mumford. Red book of varieties and schemes.
6. I. Shafarevich, Basic Algebraic Geometry.