Geometry of schemes

A.Y. 2020/2021
Overall hours
Learning objectives
The course provides an introduction to the general theory of schemes
and their main properties. In the advanced part, we expose the
students to some advanced topics including coherent and quasi-coherent
sheaves and some rudiments of birational geometry
Expected learning outcomes
The students will acquire some basic expertees that should allow them
to approach some research subjects, such as the geometry of moduli
Course syllabus and organization

Single session

Lesson period
First semester
The lessons will be on line for all the students (streaming and recorded).
Prerequisites for admission
We assume a basic knowledge of commutativa algebra (ring theory, localization of rings and modules over a ring).
Assessment methods and Criteria
The final examination consists of an oral exam. The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve some exercises.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination
Geometria degli schemi (prima parte)
Course syllabus
The course aims at giving an introduction to the theory of schemes. A scheme is a vast algebraic generalization of the concept of topological variety and allows to deal with objects which are apparently very different. For example, the affine line over the complex numbers or (the spectrum) of the ring of integers Z are very similar from the point of view of schemes. We will introduce the notion of scheme, of sheaf on a scheme and of morphism of schemes with plenty of examples. We will then study the cohomology of a sheaf on a scheme and its main properties.
We will try to be as much as possible selfcontained. In particular, we will recall the besic definitions and results from commutative algebra which are needed.
Teaching methods
Lectures in presence treating theoretical subjects, exercises and examples.
Teaching Resources
R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp.
Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, 6. Oxford Science Publications. Oxford University Press, Oxford, 2002. xvi+576 pp.
Geometria degli schemi (mod/02)
Course syllabus
In the first part of this course we will continue with the basics on the theory of schemes. In particular, we will see the following concepts: coherent sheaves, sheaf cohomology, sheaves of differentials, and smooth morphisms.
In the second part we will move to "more geometric" topics such as: divisors and invertible sheaves, projective morphisms, and blow-ups.
Teaching methods
Lectures in presence treating theoretical subjects, exercises and examples.
Teaching Resources
-R. Hartshorne, Algebraic geometry, Springer, 1977.
-Q. Liu, Algebraic geometry and arithmetic curves, Oxford University Press, 2002.
-O. Debarre, Higher-dimensional algebraic geometry. Springer, 2001.
Geometria degli schemi (mod/02)
MAT/03 - GEOMETRY - University credits: 3
Practicals: 12 hours
Lessons: 14 hours
Professor: Tasin Luca
Geometria degli schemi (prima parte)
MAT/03 - GEOMETRY - University credits: 6
Practicals: 12 hours
Lessons: 35 hours
Professor: Stellari Paolo
Fix an appointment by email
Dipartimento di Matematica "F. Enriques" - Room 2046