Geometria degli schemi
A.Y. 2025/2026
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
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
spaces.
to approach some research subjects, such as the geometry of moduli
spaces.
Lesson period: First semester
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
The first part (6 cfu) 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 notions 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 self-contained. In particular, we will recall the basic definitions and results from commutative algebra which are needed. In particular, the students are not required to attend a course about commutative algebra before. Nevertheless, it could be a good idea to attend to course on commutative algebra during the first semester of the first year of the Laurea Magistrale and the course on the geometry of schemes during the fist semester of the second year.
We will try to be as much as possible self-contained. In particular, we will recall the basic definitions and results from commutative algebra which are needed. In particular, the students are not required to attend a course about commutative algebra before. Nevertheless, it could be a good idea to attend to course on commutative algebra during the first semester of the first year of the Laurea Magistrale and the course on the geometry of schemes during the fist semester of the second year.
Prerequisites for admission
No specific prerequisites.
Teaching methods
Frontal lectures.
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.
Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, 6. Oxford Science Publications. Oxford University Press, Oxford, 2002. xvi+576 pp.
Assessment methods and Criteria
The exam consists of an oral exam during which the student will be asked to illustrate some contents of the program of the course in order to evaluate her/his understanding of the contents of the course and her/his ability in applying them by solving some exercises.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/03 - GEOMETRY - University credits: 9
Practicals: 24 hours
Lessons: 49 hours
Lessons: 49 hours
Professors:
Pertusi Laura, Stellari Paolo
Professor(s)
Reception:
Fix an appointment by email
Dipartimento di Matematica "F. Enriques" - Room 2046