Birational geometry of moduli spaces of perverse coherent sheaves on blow-ups

In order to study the wall-crossing formula of Donaldson type invariants on the blown-up plane, Nakajima–Yoshioka constructed a sequence of blow-up/blow-down diagrams connecting the moduli space of torsion free framed sheaves on projective plane, and that on its blow-up. In this paper, we prove that Nakajima–Yoshioka’s diagram realizes the minimal model program. Furthermore, we obtain a fully-faithful embedding between the derived categories of these moduli spaces.

ON THE FIRST STEPS OF THE MINIMAL MODEL PROGRAM FOR THE MODULI SPACE OF STABLE POINTED CURVES

The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation, and we study their geometric properties. As a particular case, we recover the first few Hassett–Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.

New directions in the Minimal Model Program

We survey some recents developments in the Minimal Model Program. After an elementary introduction to the program, we focus on its generalisations to the category of foliated varieties and the category of varieties defined over any algebraically closed field of positive characteristic.

Smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$

International audience We classify all smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.

Higher-dimensional foliated Mori theory

We develop some foundational results in a higher-dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman–Mori cone of curves in terms of the numerical properties of $K_{{\mathcal{F}}}$ for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program (MMP) for rank 2 foliations on threefolds.