Second Level Degree in Mathematics
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ALGEBRAIC GEOMETRY 2 - 6 CFU
Teacher
Carla Novelli
Scheduled Period
I Year - 1 Semester | 30/09/2019 - 18/01/2020
Hours: 48 (24 esercitazione, 24 lezione)
Prerequisites
Basics on topology and commutative algebra.
Target skills and knowledge
A good knowledge of the algebraic objects used in Birational Geometry.
Examination methods
Seminar.
Assessment criteria
The student will be evaluated on his/her understanding of the topics, on the acquisition of concepts and methodologies proposed and on the ability to apply them in full independence and awareness.
Course contents
Introduction to affine and projective varieties.
Morphisms, rational maps and birational maps.
Singularities and resolution of singularities. Blow-ups.
Introduction to sheaves and cohomology.
Rational curves and divisors on varieties.
Ampleness and cones of curves.
Extremal rays and extremal contractions.
Surfaces: Cone Theorem, birational classification and Minimal Model Program.
Higher dimensional varieties: Cone Theorem, Contraction Theorem, Extremal Rays, contractions associated with extremal rays, introduction to Minimal Model Program and Minimal Models.
Planned learning activities and teaching methods
Lectures and recommended exercises.
Additional notes about suggested reading
Further material will be available in the moodle page of the course.
Textbooks (and optional supplementary readings)
- Olivier Debarre, Higher-Dimensional Algebraic Geometry, New York, Universitext, Springer-Verlag, 2001.
- Arnaud Beauville, Complex Algebraic Surfaces (Second Edition), London Mathematical Society., Cambridge: Cambridge University Press, 1996. Student Text 34
- Ja'nos Kolla'r & Shigefumi Mori, Birational Geometry of Algebraic Varieties, Cambridge, Cambridge University Press, 1998. Cambridge Tracts in Mathematics 134
- Kenji Matsuki, Introduction to the Mori Program, New York, Universitext, Springer-Verlag, 2002.