Geometry of schemes

Academic year

First semester of the academic year 2023-2023 (6+3 cfu).

Office hours

You can fix an appointment by sending an email to Paolo Stellari for the first part of the course or to Laura Pertusi for the second part.


Teaching will be in presence according to the following schedule:.

  • Tuesday 10:30-12:30.
  • Wednesday 13:30-15:30.
  • Thursday 8:30-10:30.


No lectures during the week November 6-10, 2023.

For information on how to attend the lectures, please consult Ariel.


The course consists of two parts.

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.

In the second part of this course (3 cfu) 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.


  • First Homework (A.A. 2023-2024): pdf.
  • Second Homework (A.A. 2023-2024): pdf.


  • O. Debarre, Higher-dimensional algebraic geometry, Springer, 2001.
  • 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.