Lectures and schedule
Lecture 1: 12/02/2021, 10:00 - 13:00
Introduction and overview of the program of this course. Review of some preliminaries topics as divisors, line bundles, curves, Riemann-Roch theorem.
Some material:
210212_introduzione.pdf
210212_materiale.pdf
Exercises:
Lecture 2: 19/02/2021, 10:00 - 13:00
Geometric Riemann-Roch theorem, Clifford theorem, Castelnuovo theorem.
Material:
Exercises:
Lecture 3: 26/02/2021, 10:00 - 13:00
Coverings, Riemann-Hurwitz and Riemann’s existence theorem.
Rational normal scrolls.
Extremal Castelnuovo curves.
Gonality. Geometric menaning of gonality: the gonal scroll.
Lecture 4: 5/3/2021, 10:00 - 13:00
Quadrics through a canonical curve: Enriques-Babbage theorem.
The Maroni invariant for trigonal curves.
Lecture 5: 12/3/2021, 10:00 - 13:00
Gonality and coverings
Gonality and Abel Jacobi map
Clifford index. Basic properties.
Curves with low Clifford index.
Introduction to the Green conjecture.
Lecture 6: 26/3/2021, 10:00 - 13:00
Introduction to abelian varieties and to the Albanese morphism
Preliminaries on algebraic surfaces
First inequalities for irregular surfaces
Lecture 7: 9/4/2021, 10:00 - 13:00
Fibrazioni: Castelnuovo de Franchis. Generalità sulle superfici. Invarianti. Teorema di Noether. Contraibilità di Castelnuovo. Idee sulla classificazione.
Disuguaglianza di Noether. Positività degli invarianti nel caso di tipo generale. Ancora fibrazioni: invarianti relativi e positività di Arakelov e di Kollar. Essere isotriviale vs essere localmente banale (Serrano).
Lecture 8: 16/4/2021, 10:00 - 13:00
Metodo di Xiao. Disuguaglianze slope. Corollario se K_f^2=0 allora f è localmente banale. Raffinamenti: il caso trigonale e il caso di irregolarità relativa positiva. Questioni aperte. Metodo di Lu e Zuo.
Lecture 9: 23/4/2021, 10:00 - 13:00
Appendice sui line bundles sulle varietà abeliane.
La dimostrazione di Pardini della disuguaglianza di Severi per superfici di tipo generale e di dimensione di Albanese massima.
Lecture 10: 30/4/2021, 10:00 - 13:00
Bogomolov–Miyaoka–Yau inequality
Seminar: 7/5/2021, 10:00-13:00. Speaker: Pietro Pirola
Title: Del teorema di Xiao per fibrazioni su base razionale, della congettura di Xiao e del suo controesempio.
Seminar: 18/5/2021, 14:00-16:00. Speaker: Daniel Greb
Title: Characterising projective space via stability and a Chern class condition
Abstract: After quickly discussing classical uniformisation results for Kähler-Einstein projective manifolds with ample, torsion or anti-ample canonical bundle, I will explain some sheaf-theoretic techniques that allow to characterise projective spaces among Fano manifolds satifying a certain algebrogeometric stability condition. While the discussion will be aimed towards graduate students, towards the end I will shortly discuss how the techniques generalise to the case of varieties with singularities as they appear in the minimal model program and with nef anti-canonical bundle.