
Course unit
ALGEBRAIC GEOMETRY 2
SC02120637, A.A. 2017/18
Information concerning the students who enrolled in A.Y. 2017/18
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/03 
Geometry 
6.0 
Mode of delivery (when and how)
Period 
Second semester 
Year 
1st Year 
Teaching method 
frontal 
Organisation of didactics
Type of hours 
Credits 
Hours of teaching 
Hours of Individual study 
Shifts 
Practice 
3.0 
24 
51.0 
No turn 
Lecture 
3.0 
24 
51.0 
No turn 
Start of activities 
26/02/2018 
End of activities 
01/06/2018 
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 unit contents:

Introduction to affine and projective varieties.
Morphisms, rational maps and birational maps.
Singularities and resolution of singularities. Blowups.
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) 

Arnaud Beauville, Complex Algebraic Surfaces (Second Edition). London Mathematical Society.: Cambridge: Cambridge University Press, 1996. Student Text 34

Olivier Debarre, HigherDimensional Algebraic Geometry. New York: Universitext, SpringerVerlag, 2001.

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, SpringerVerlag, 2002.


