401-4148-17L Moduli of Maps and Gromov-Witten invariants
|Semester||Spring Semester 2017|
|Language of instruction||English|
|Abstract||Enumerative questions motivated the development of algebraic geometry for centuries. This course is a short tour to some ideas which have revolutionised enumerative geometry in the last 30 years: stable maps, Gromov-Witten invariants and quantum cohomology.|
|Objective||The aim of the course is to understand the concept of stable maps, their moduli and quantum cohomology. We prove Kontsevich's celebrated formula on the number of plane rational curves of degree d passing through 3d-1 given points in general position.|
1) Brief survey on moduli spaces: fine and coarse moduli.
2) Stable n-pointed curves
3) Stable maps
4) Enumerative geometry via stable maps
5) Gromov-Witten invariants
6) Quantum cohomology and quantum product
7) Kontsevich's formula
|Literature||The main reference for the course is:|
J. Kock and I.Vainsencher: Kontsevich's Formula for Rational Plane Curves
-Algebraic varieties: I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
-Moduli of curves: Joe Harris and Ian Morrison, Moduli of Curves, Graduate Texts in Mathematics 187, Springer-Verlag
-Moduli spaces (fine and coarse): Peter. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes, Springer-Verlag
|Prerequisites / Notice||Some minimal background in algebraic geometry (varieties, line bundles, Grassmannians, curves).|
Basic concepts of moduli spaces (fine and coarse) and group actions will be explained mainly through examples.