Algebraic Varieties and their Moduli

25 May 2015 - 5 June 2015

Research Directions

The aim of this intensive period is to investigate some geometric aspects of algebraic varietis in any dimension, with particular attention to their moduli spaces. This first week is devoted mainly to graduate students and postdoctoral fellows, but also to anybody else interested in the subject. It is actually a school aimed at recovering some background, with a couple of talks by young participants. The attention will be indicatively focused on Enriques-Kodaira classification of algebraic surfaces, K3 surfaces, rational curves on projective varieties, irreducibile symplectc varieties, stable curves and deformation theory. The lecturers will be M. Andreatta, G. Bini, D. Iacono, G. Mongardi, F. Polizzi, R. Pignatelli.
They will talk about the following subjects.

• Marco Andreatta (Università di Trento). "Rational curves on Projective Varieties" (Course text)
• Gilberto Bini (Università di Milano). "Chow Groups of Projective Varieties"
• Donatella Iacono (Università di Bari). "Some Deformation Theory"
• Roberto Pignatelli (Università di Trento). "Surfaces of general type" (Course text)
• Francesco Polizzi (Università della Calabria). "K3 Surfaces"
• Giovanni Mongardi (Universität Bonn) "Hyperkähler Manifolds"

The second week is focused on the following three courses:

• Enrico Arbarello (“Sapienza” Università di Roma) "Curves and Sheaves on K3 Surfaces"
  - The geometry of curves on K3 surfaces.
  - Moduli of sheaves on K3 surfaces
• Gianluca Pacienza (IRMA Université de Strasbourg) "Rational Curves and Chow Groups of Irreducible Symplectic Varieties"
  - Mumford’s theorem on 0-cycles and its generalizations.
  - The Chow ring of K3 surfaces.
  - Rational curves and Chow groups of irreducible symplectic varieties.
  - The Bloch-Beilinson filtration and coisotropic families of rational curves.
• Soenke Rollenske (Universität Marburg) "An Introduction to Stable Surfaces"
  - Introduction to semi-log-canonical surface singularities.
  - Definition of the moduli space of stable surfaces and first examples (topological and holomorphic invariants).
  - Extension of Bombieri's result for pluricanonical maps and geography.
  - Special families of surfaces, e.g., Godeaux surfaces.

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This research period is organized with the partecipation and support of the following institutions:
- INDAM.
- Foundation Compositio Mathematica.
- Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano.
Firb "Spazi di Moduli e Applicazioni".
-Dipartimento di Matematica e Fisica, Università degli Studi di Roma Tre.
Firb "Spazi di Moduli e Applicazioni".
- Dipartimento di Matematica, Università degli Studi di Trento.
Firb "Spazi di Moduli e Applicazioni".
- Dipartimento di Matematica "G. Castelnuovo", Sapienza Università di Roma.
Firb "Prospettive in Teoria di Lie".
- Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano.
Prin "Geometria delle varietà algebriche".
- Dipartimento di Matematica "Leonida Tonelli", Università degli Studi di Pisa.
Prin "Geometria delle varietà algebriche".
- Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata
Prin "Geometria delle varietà algebriche".
- Dipartimento di Matematica, Università degli Studi di Bari "Aldo Moro".
Gruppo di “Geometria e Algebra”.
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