Inequalities between invariants of varieties

This is the official page of the PhD course "Inequalities between invariants of varieties", helded by Riccardo Moschetti and Lidia Stoppino.

The course is devoted to the study of various inequalities between invariants of algebraic varieties, focusing on some remarkable results like Clifford's and Castelnuovo's inequalities for curves, the Bogomolov-Miyaoka-Yau inequality, and the Cornalba-Harris-Xiao's inequalitiy for surfaces.

Prerequisites are some basic notions about the theory of algebraic varieties, line bundles, maps in projective spaces and some rudiments on abelian varieties.

When: The course will take places roughly from mid-February to the end of March 2021 for approximately 27 hours. In addition to the lectures, there will be some seminars on recent developments and new generalisation of classical results treated in the course. Such seminars could take place also in April and May, depending on the availability of the speakers.

Where: The course will be held online via Zoom. We ask the participants to write us an e-mail, so we can share the link and the password of the meetings.

Abstract and references: In the first part of the course we will talk about curves, treating Riemann Roch theorem, its geometric version, and gonality. We will discuss special divisor on curves: Clifford's theorem and Castelnuovo's bound. We will then discuss Clifford's index and its relation with gonality, and give an overview of Green's conjecture \\

Then we will move to surfaces, introducing first some basic notions about invariants, Hodge index theorem, fibred surfaces, and relative invariants, and then studying Bogomolov-Miyaoka-Yau inequality. We will prove Cornalba-Harris-Xiao's inequalities for fibred surfaces and prove some generalisations. We will end by showing the Pardini-Severi inequality for surfaces of maximal Albanese dimension.

These are the basic reference books or survey papers of the course. During the lectures also some papers will be proposed to the participants.

  • Arbarello, Cornalba, Griffiths, Harris, Geometry of algebraic curves. Vol. I.

  • Arbarello, Cornalba, Griffiths, Geometry of algebraic curves. Vol. II.

  • Barth, Hulek, Peters, Van de Ven, Compact complex surfaces.

  • Beauville, Complex algebraic surfaces.

  • Griffiths, Harris, Principles of algebraic geometry.

  • Lopes, Pardini, The geography of irregular surfaces.

  • Miranda, Algebraic curves and Riemann surfaces.

  • Reid, Chapters on Algebraic Surfaces.