Knowledge and understanding: at the end of the course the student is
required to prove that he/she knows the basic tools and results of
complex algebraic geometry.

Applying knowledge and understanding: at the end of the course the
student must know how to apply the material of the course to study the
geometric properties of the set of solutions of a system of polynomial
equations.

Making judgments: at the end of the course the student will know how to
recognize and how to apply the acquired techniques of algebraic
geometry, and will also recognize the situations and problems in which
these techniques can be advantageously used.

Communication skills: at the end of the course the student will be able to express himself/herself appropriately on the above topics.

Learning skills: at the end of the course the student will be able to consult
standard manuals
of basic algebraic geometry.

Basic familiarity with linear algebra, affine and projective geometry,
topology, real and complex analysis, differential geometry,
at the level of a Laurea in Matematica program.

Linear algebra, affine and projective spaces and their subspaces; basic
notions of general topology; a basic knowledge of plane algebraic curves
is useful but not essential.

Affine and algebraic varieties. Dimension theory. Local properties. Projective varieties. Examples. Elimination theory. Morphisms.

G. R. Kempf, "Algebraic Varieties", London Mathematical Society, Lecture Notes Series 172 D. Mumford, "Algebraic Geometry I, Complex Projective Varieties", Springer Berlin, Heidelberg, 1995.

DEFINITION OF ALGEBRAIC VARIETIES. GENERAL CONSTRUCTION OF AFFINE VARIETIES. HILBERT'S NULLSTELLENSATZ. SUBVARIETIES. NOETHERIANITY. IRREDUCIBILITY. DIMENSION THEORY. HYPERSURFACES AND THE PRINCIPAL IDEAL THEOREM. PRODUCTS. SEGRE EMBEDDING. SEPARATEDNESS. PROJECTIVE NULLSTELLENSATZ. ELIMINATION THEORY. PROJECTIONS AND THE NOETHER NORMALIZATION LEMMA. SMOOTHNESS.

Lessons and exercises sessions. During the course some exercises will be
assigned as homework, their solutions will be discussed in class.

There will be a moodle page of the course in which it will be reported: the
diary of the lessons; the dates of the exams; and additional material.

The final exam is aimed at ascertaining the knowledge of the topics of
the entire program of the course, expression skills and language skills of
the students. It consists of an oral exam.