**Abstracts:**

**The Lectures will be in english.**

**FIRST WEEK:**

**1. Algebraic Curves and their moduli spaces **

*Classical approach (Edoardo Sernesi)*

**Abstract:** The classification problem. The notion of curve with general moduli. Some important families of curves. Non-trivial results without the definition of moduli. Brill-Noether, Severi, Petri. The classical vs. the functorial point of view . The modern approach. Some classical problems that are still open.

**2. Higher dimensional varieties and their moduli spaces **

**(Oliver Debarre)**

**Abstract:** In this course, I will try to demonstrate the importance of rational curves in birational geometry. I will in particular explain Mori's original approach to the Cone Theorem, which is based on the construction of rational curves on smooth varieties whose canonical bundle is negative, via his famous bend-and-break lemmas.

1) Rational curves

2) The Cone Theorem

3) Contractions

3.1 Relative cone of curves

3.2 Contraction of an extremal subcone

4) Parametrizing morphisms

4.1 The scheme Mor(Y,X)

4.2 The tangent space to Mor(Y,X)

4.3 The local structure of Mor(Y,X)

4.4 Morphisms with fixed points, flat families

4.5 Morphisms from a curve

5) Producing rational curves

6) Rational curves on Fano varieties

7) A stronger bend-and-break lemma

8) Rational curves on varieties whose canonical bundle is not nef

9) Proof of the Cone Theorem

9.1 Elementary properties of cones

9.2 Proof of the Cone Theorem

10) The Cone Theorem for projective klt pairs

10.1 The Rationality Theorem

10.2 Proof of the Cone Theorem in the singular case

10.3 Existence of contractions

**3. Geometric Invariant Theory and Bridgeland stability **

*Lectures on GIT and moduli (Radu Laza *)*

**Abstract: **Geometric Invariant Theory (GIT) is an important tool in the study of moduli spaces in algebraic geometry. In these lectures we will review the basic construction and properties of GIT quotients. We will also discuss some of the more recent developments including variation of GIT quotients (VGIT), the connections between GIT/VGIT and birational geometry, and the related notion of K-stability and the relationship to the existence of special metrics. We will close by reviewing some classical as well as more recent applications of GIT to moduli problems.

Standard References:

[GIT] D. Mumford et al., Geometric invariant theory, third ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, Springer-Verlag, Berlin, 1994.

[Dol] I. V. Dolgachev, Lectures on invariant theory , London Mathematical Society Lecture Note Series, vol. 296, Cambridge University Press, Cambridge, 2003.

[Muk] S. Mukai, An introduction to invariants and moduli , Cambridge Studies in Advanced Mathematics, vol. 81, Cambridge University Press, Cambridge, 2003.

Surveys related to the lectures:

[Laz1] GIT and moduli with a twist, in "Handbook of Moduli" vol. 2, Adv. Lect. Math. 25 (2013), Int. Press, 259-297.

[Laz2] Perspectives on the construction and compactification of moduli spaces, Lectures for the school on "Compactifying Moduli Spaces" (Barcelona, May 2013), to appear in a volume of CRM Lecture Notes.

**4. Algebraic Curves and their moduli spaces**

*Syzygies of algebraic curves (Gabi Farkas) *

**Abstract:** The main theme of these lectures is to present recent progress on syzygies of algebraic curves using geometric methods. The first lectures will deal with the basics of minimal free resolutions and Koszul cohomology of projective varieties. Then we will explain the statements of the Green respectively Green-Lazarsfeld secant conjectures. Voisin's solution to the generic Green conjecture will be sketched. Finally we will discuss generalizations of Green's conjecture to paracanonical curves and sketch a solution using special K3 surfaces.

Recommended bibliography:

1. M. Aprodu and J. Nagel: Koszul cohomology and algebraic geometry,

University Lecture Series AMS.

2. M. Aprodu and G. Farkas: Koszul cohomology and applications to moduli,

Clay Math Proceedings, http://arxiv.org/abs/0811.3117

3. A. Beauville: La conjecture Green generique,

Seminar Bourbaki, http://arxiv.org/abs/math/0311471

4. G. Farkas and M. Kemeny: The generic Green-Lazarsfeld Secant Conjecture,

Inventiones Math. 2016, http://arxiv.org/abs/1408.4164

5. M. Green and R. Lazarsfeld: On the projective normality of complete

linear series on algebraic curves, Inventiones Math. 1986.

**SECOND WEEK**

**1. Higher dimensional varieties and their moduli spaces **

*Higher dimensional varieties and their moduli spaces (Paolo Cascini) *

**Abstract: **The goal of the Minimal Model Programme, started by S. Mori in the 1980s, is to generalise the main results of the classification of algebraic surfaces, due to Castelnuovo, Enriques and Severi, to higher dimensional projective varieties. In particular, it predicts the existence of a birational model for any complex projective variety which is as simple as possible. The last decade has seen exceptional activity towards Mori's programme and its applications.

The goal of these lectures is to discuss some recent results, applications and new aspects of this programme.

More specifically, we plan to cover the following topics:

**Lecture 1:** Review of the Minimal Model Program.

Vanishing theorems and lifting theorems. In particular, we will describe a new approach to lift sections. **Lecture 2: ** Finite generation of the canonical ring (old and new point of view). Existence of flips. **Lecture 3: **We will describe some applications and some open problems related to these results. In particular we will describe positive characteristic methods and analytic methods in birational geometry. **Lecture 4: **We will introduce Shokurov's log geography and some of its main applications: Sarkisov programme and termination of flips. **Lectures 5: **ACC and boundedness results: towards the moduli space of varieties of general type.

**2. Geometric Invariant Theory and Bridgeland stability**

*Introduction to Bridgeland stability (Emanuele Macri)*

**Abstract: **In this series of lectures we will cover the following topics:

(1) Basic definitions of stability conditions on derived categories.

(2) Moduli spaces of stable objects and variation of stability.

(3) The case of surfaces.

(4) The higher dimensional case and open problems.

**3. Minimal Model Program/Birrational geometry and Topology of Mg.**

*Birrational geometry and Topology of Mg. (Samuel Grushevsky)*

**Abstract: **In the second week of the course we will continue to investigate the study of the topology and birational geometry of the moduli space of curves, focusing on highlighting the similarities and differences with the moduli spaces of abelian varieties, and especially comparing the techniques available for the study of both, in low genus, and in general.

**4. Tropical geometry and applications to moduli and degenerations: (Melody Chan)**

*Moduli and degenerations of algebraic curves via tropical geometry*

**Abstract: **Tropical geometry is a modern degeneration technique in algebraic geometry, bringing combinatorics and nonarchimedean geometry to the study of algebro-geometric objects. The focus of these lectures will be tropical curves and their moduli spaces. We will cover, time permitting:

- tropical curves; tropicalization of algebraic curves
- toroidal compactifications; Deligne-Mumford compactification by stable curves; tropicalization thereof
- applications to topology of moduli spaces
- recent directions in algebraic and tropical Brill-Noether theory