Geometry of schemes

Academic year

First semester of the academic year 2018-2019.

Office hours

You can fix an appointment by sending an email.


  • Monday 10:30-12:30 in Room 5.
  • Tuesday 8:30-10:30 in Room 3.


The lectures will start on October 1. During the week October 29-November 2 there are no lectures.


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 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.

Syllabus: pdf.


  • First exercise sheet: pdf.
  • Second exercise sheet: pdf.


  • 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.