Research Seminar "Geometric Introduction to Algebraic Geometry"
- Students will be competent in basic constructions of algebraic geometry that will allow them to start studying more advanced courses in algebraic geometry and apply their knowledge to study the courses like algebraic groups, algebraic curves and surfaces.
- Students will learn basic constructions, theorems of algebraic geometry. Also they will gain sufficient package of examples of algebraic varieties and the method of their study.
- Projective spaces. Geometry of projective quadrics. Spaces of quadrics.
- Lines, conics, and PGL(2). Rational curves and Veronese curves. Plane cubic curves.
- Grassmannians, Veronese's, and Segre's varieties. Examples of projective maps coming from tensor algebra.
- Elements of commutative algebra: Integer elements in ring extensions, finitely generated algebras over a field, transcendence generators, Hilbert's theorems.
- Affine Algebraic Geometry - Commutative Algebra dictionary. Maximal spectrum, pullback morphisms, Zariski topology, geometry of ring homomorphisms.
- Agebraic manifolds, separateness. Irreducible decomposition. Projective manifolds, properness. Rational functions and maps.
- Dimension. Dimensions of subvarieties and fibers of regular maps. Dimensions of projective varieties.
- Vector bundles and their sheaves of sections. Vector bundles on the projective line. Linear systems, invertible sheaves, and divisors. The Picard group.
- Solutions of problems from home task sheetsWritten solutions
- Final examWritten exam
- Interim assessment (2 module)0.5 * Final exam + 0.5 * Solutions of problems from home task sheets
- Alexey L. Gorodentsev. Algebra II: Textbook for Students of Mathematics. Springer, 2017. ISBN: 9783319508535,3319508539
- Igor R. Shafarevich Basic Algebraic Geometry 1 Varieties in Projective Space, Springer, 2013.
- Igor R. Shafarevich. Basic Algebraic Geometry 2 Schemes and Complex Manifolds, Springer, 2013