Cremona transformations and the Sarkisov program

Alberto Calabri
Title: Recent results on plane Cremona transformations
Abstract: The group of plane Cremona transformations is endowed with a natural topology and the set of maps of some bounded degree is closed. We will review some known properties of the quasi-projective variety which parametrizes plane Cremona maps of fixed degree. Then, we will address the question of determining which plane Cremona maps of small degree are limits of maps of higher degree. This talk is mainly based on joint works with Jérémy Blanc and with Cinzia Bisi and Massimiliano Mella.

Anne-Sophie Kaloghiros
Title: Volume preserving maps of Calabi-Yau pairs with a toric model
Abstract: A Calabi-Yau pair (X, D) consists of a normal projective variety X and a reduced integral divisor D with K_X + D = 0. A birational map (X, D_X ) ---> (Y, D_Y) between CY pairs is volume preserving if pullbacks of K_X + D_X and K_Y + D_Y to high enough models coincide.
The pair formed by a toric variety and its boundary divisor is an example of CY pair, and mutations of algebraic tori can be extended to volume preserving birational maps of toric pairs. In this talk, I will discuss a conjecture that states that every volume preserving birational map between CY pairs that have a toric model (i.e. that are volume preserving birational to toric pairs) is a chain of mutations. I will give some evidence in support of the conjecture in dimension three.

Giovanni Staglianò
Title: Kuznetsov's conjecture and rationality of cubic fourfolds
Abstract: We recall the main conjectures about the important classical problem (still unsolved) on the rationality of smooth cubic hypersurfaces in a 5-dimensional projective space, cubic fourfolds for short, and we present recent contributions in favor of these conjectures. We will also briefly illustrate similar conjectures and results for Gushel-Mukai fourfolds, that is for smooth quadric hypersurfaces in del Pezzo fivefolds. The talk is based on some joint works with Francesco Russo and the recent collaboration with Michael Hoff.