Mohammad Ehtisham Akhtar _ Institut des Hautes Études Scientifiques (IHES)
Mutations and the Classification of Fano Varieties
The classification of Fano varieties is an important long-standing problem in algebraic geometry. Mirror symmetry predicts that this problem should be equivalent to classifying a suitable class of Laurent polynomials up to an appropriate notion of equivalence. Recent work of Coates, Corti, Golyshev et al. suggests that the correct equivalence relation to impose is algebraic mutation of Laurent polynomials. These lectures will introduce algebraic mutations and discuss the notion of combinatorial mutations, which are transformations of lattice polytopes induced by algebraic mutations of Laurent polynomials supported on them. Our focus will be on the case of surfaces, where the theory is particularly rich. Particular attention will be given to the role played by combinatorial mutations in the classification of Fano orbifold surfaces.
Thomas Ducat _ Research Institute for Mathematical Sciences (RIMS)
Divisorial extractions from singular curves in a smooth 3-fold
I will introduce the Kustin--Miller unprojection theorem and explain how serial unprojection can be used to construct divisorial extractions from a singular curve C contained in a smooth 3-fold X. By Reid's general elephant conjecture, an anticanonical section containing C should have at worst Du Val singularities. Then we can study such extractions according to the ADE types that describe Du Val singularities. The type A extractions lead to a class of affine Gorenstein 4-folds which are very similar to Brown and Reid's `diptych varieties'.
A very rough guide to what will be covered:
Lecture 1 - Uniqueness of 3-fold divisorial extractions from a curve. The general elephant conjecture. Introduction to Kustin--Miller unprojection.
Lecture 2 - Examples in low codimension: Prokhorov and Reid's construction, type D and E cases.
Lecture 3 - Examples in high codimension: type A cases. Type A extractions as a QQ-smoothing of a singular surface. Extractions with an irreducible central fibre.
Andreas Krug _ Universität Marburg
Derived categories of coherent sheaves and their autoequivalences
The aim of these lectures is to explain some aspects of the theory of derived categories of coherent sheaves on smooth varieties. The emphasis will be on the study of autoequivalences of these categories. We will introduce general methods for the construction of autoequivalences. The examples that we are going to discuss will be in the context of the McKay correspondence and Hilbert schemes of points on surfaces.
Patrick Popescu-Pampu _ Université Lille 1
Topology of the Milnor fibers of sandwiched surface singularities
Sandwiched surface singularities form a special class of rational surface singularities, introduced by Hironaka and Spivakovsky during their study of Semple-Nash modifications of arbitrary surface singularities. Being rational, they are smoothable. Their Milnor fibers are by definition the generic fibers of all of their smoothings. De Jong and Van Straten showed that their Milnor fibers may be studied by deforming in special ways associated weighted plane curve singularities. Némethi and myself showed how to extract from the topology of the Milnor fibers the main combinatorial object used by them to manipulate those deformations. My lectures will be an introduction to those various works.
Jinhyung Park _ Korea Institute for Advanced Study
Cox rings of rational surfaces
Cox rings were introduced by Hu and Keel to generalize constructions of toric varieties due to Cox. Cox rings decode lots of information of birational structures and arithmetic properties of Mori dream spaces. The study of Cox rings of rational surfaces is already a very interesting problem. In these lectures, I first give a brief survey of Cox rings and Mori dream spaces, and then review results on the finite generation of Cox rings of rational surfaces, the classification of Mori dream rational surfaces via redundant blow-ups, and syzygies of Cox rings of del Pezzo surfaces