PROJECTIVE GEOMETRY
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- Versione italiana
- Academic year
- 2019/2020
- Teacher
- ALEX MASSARENTI
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- MAT/03
Training objectives
- Toric varieties provide an elementary way to see many examples and phenomena in algebraic geometry.
The goal of the course is an introduction to toric varieties and to their combinatorial, topological and geometric properties.
At the end of the course, the student will be able to recognize and construct examples of toric varieties and to describe their properties. Prerequisites
- Geometria I, II and III of the three-year laurea in Mathematics.
It is useful to have already attended the course of Geometria Algebrica I, even if it is not strictly necessary. Course programme
- Convex polyhedral cones (2 hours).
Affine toric varieties (2 hours).
Fans and toric varieties (2 hours).
Toric varieties from polytopes (2 hours).
Local properties of toric varieties (2 hours).
Toric surfaces (2 hours).
Quotient singularities (2 hours).
One-parameter subgroups and limit points (2 hours).
Compactness and properness (2 hours).
Non-singular surfaces (2 hours).
Resolution of singularities (2 hours)
The orbit-cone correspondence (2 hours)
Fundamental groups and Euler characteristics (2 hours)
Divisors (2 hours)
Line bundles (2 hours)
Cohomology of line bundles (2 hours)
Chow groups (2 hours)
Cohomology of smooth toric varieties (2 hours)
Riemann-Roch theorem (2 hours)
Bézout theorem (2 hours)
Serre duality (2 hours) Didactic methods
- Frontal lessons on the blackboard.
Learning assessment procedures
- The exam is oral. The student will be asked to describe the geometric properties of a toric variety defined by a polytope or a fan.
Reference texts
- W. Fulton,
Introduction to toric varieties,
Annals of Mathematics Studies, 131.
The William H. Roever Lectures in Geometry.
Princeton University Press, Princeton, NJ, 1993.
D.A. Cox, J.B. Little, H.K. Schenck,
Toric varieties,
Graduate Studies in Mathematics, 124.
American Mathematical Society, Providence, RI, 2011.
