Geometry of Schemes (first part)
A.Y. 2019/2020
Learning objectives
The course provide an introduction to the general theory of schemes and their main properties.
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.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
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 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.
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.
Prerequisites for admission
We assume a basic knowledge of commutativa algebra (ring theory, localization of rings and modules over a ring).
Teaching methods
Lectures at the blackboard which include the explanation of theoretical notions, 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.
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 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.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/03 - GEOMETRY - University credits: 6
Lessons: 42 hours
Professor:
Stellari Paolo
Shifts:
-
Professor:
Stellari PaoloProfessor(s)
Reception:
Fix an appointment by email
Dipartimento di Matematica "F. Enriques" - Room 2046