Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
Academic year in which the course will be held: 
Course type: 
Compulsory subjects, characteristic of the class
First Semester
Standard lectures hours: 
Detail of lecture’s hours: 
Lesson (64 hours)

Basic courses (first two years of undergraduate studies) in analysis topology and geometry

Final Examination: 

The exam will be oral, a part of it will be the exposition of the solution of one or more of the exercises from a list given during the course.

Voto Finale

Provide the student with an introduction to the modern language of algebraic geometry, through a discussion of the basic results of Riemann's classical theory of surfaces. In this theory, concepts and results come from algebraic topology, from differential geometry, from complex analysis, from commutative algebra, from algebraic geometry of manifolds or from projective or abstract schemes.

Riemann Surfaces (SdR), basic definitions: complex maps and complex structures. First examples. Recalls on the classification of compact surfaces. Genus of a compact SdR.
Complex projective spaces.
Affine and projective algebraic curves. Associated SDR.
Holomorphic and meromorphic functions on a SdR.
Holomorphic maps between SdR. Multiplicity of a point.
The sum of the orders of a meromorphic function is zero.
Maps between compact SDRs. Degree.
Euler characteristic and triangulations. Hurwitz theorem (topological proof).
Automorphisms and group actions on a SdR. Monodromy.
Hyperelliptic surfaces.
Integration on SdR: Differential forms. Operations on forms. Lemmas of Poincare and Dolbeaux. Stokes' theorem. Residue theorem.
Divisors and meromorphic functions.
Complex vector bundles on a SdR and to correspondence between divisors and vector bundles of rank 1.
Linear equivalence of divisors, spaces of forms and functions associated with a divisor. divisors and maps in the projective space. Very ample divisors. Algebraic curves.
Riemann Roch theorem, proof.
Applications of Riemann Roch: a SdR is an algebraic curve. Each algebraic curve is projective. Clifford's theorem. Canonical map, hyperelliptic curves. Geometric form of Riemann-Roch. Calculation of the Riemann parameters.
Degree of projective curves. Monodromy of the hyperplane divisors. Lemma of uniform position, and Lemma of general position. Bound of Castelnuovo. Maximal genus curves. Inflection points and Weierstrass points.
Homology, periods and Jacobians.
The Abel-Jacobi map. Proof of Abel's Theorem.

Bundles on curves. Introduction to Cech cohomology. Picard group of invertible sheaves and correspondence between divisors and invertible sheaves.

R. Miranda, Algebraic Curves and Complex Surfaces, Graduate Studies in Mathematics, Vol 5, 1991.
Handwritten notes of the lessons will also be available on the e-learning platform.

Frontal lectures. Exercices in classroom end homework. Student seminars

Other textbooks or journal articles can be used during the course, and they will be provided by the teacher