Monday Afternoon
Room: B17 (building 1TP1)
Tuesday Morning
Room: B17 (building 1TP1)
Tuesday Afternoon
Room: Schwartz (building 1R3)
Wednesday Morning
Room: Schwartz (building 1R3)
Wednesday Afternoon
Room: B17 (building 1TP1)
Ana-Maria Castravet, Higher Fano manifolds
Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been great effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo Brauer obstruction). In this talk, I will discuss a possible notion of higher Fano manifolds in terms of positivity of higher Chern characters, and give a survey of what is currently known.
Enrico Fatighenti, Fano varieties of K3 type and their properties
Fano varieties of K3 type are a special class of Fano varieties, which are usually studied for their link with hyperkaehler geometry, rationality properties, and much more. In this talk, we will recap some recent results, obtained jointly with Bernardara, Manivel, Mongardi, and Tanturri, that focus on the explicit construction of examples and the study of their Hodge-theoretical properties.
Michał Kapustka, Comparing double EPW sextics with double EPW cubes
Double EPW cubes and double EPW sextics are among few
known locally complete families of projective
hyperkaehler manifolds. They both arise from the same
data: a Lagrangian space in the third exterior power of
a six-dimensional space equipped with the natural
symplectic form given by wedge product. In this talk we
consider relations between these two families. In
particular, we compare their period maps and
investigate moduli spaces of minimal degree elliptic
curves on very general manifolds in both families.
As a consequence, combining our results with known
results on double EPW sextics, we prove that a very
general double EPW cube is the moduli space of stable
objects on the Kuznetsov component of its corresponding
Gushel-Mukai fourfolds; this answers a problem posed by
Perry, Pertusi and Zhao. We moreover show that the
moduli spaces of minimal degree elliptic curves on a
very general double EPW sextic and the corresponding
double EPW cube are isomorphic curves. This provides
further evidence for a conjecture of Nesterov and
Oberdieck and raises interesting questions. This is
joint work with Grzegorz Kapustka and Giovanni
Mongardi.
Alexander Kuznetsov, Derived categories of families of Fano threefolds
I will describe the structure of derived categories of smooth Fano fibrations of relative Picard rank 1 with rational geometric fibers over arbitrary bases.
Christian Pauly, Wobbly bundles on curves of low genus
In this talk I will first review the classical geometry of the moduli spaces of semi-stable vector bundles over a smooth complex projective curve. When the rank of the vector bundles and the genus of the curve is two or three, the corresponding moduli spaces can be described by hypersurfaces in projective space introduced by Coble. In the second part I will introduce the notion of very stable bundle due to Drinfeld and explain some conjectures on the locus of non-very stable or wobbly bundles. Finally I will explain some recent work by Pal relating the wobbly locus to non-free minimal rational curves.
Yulieth Prieto, Hyperkähler manifolds of K3^{[n]}—type admitting symplectic birational maps
Motivated by the existence of birational involutions on projective hyperkähler manifolds which are deformation equivalent to Hilbert schemes of n points of K3 surfaces, we show that such hyperkähler manifolds are always birational to moduli spaces of (twisted) stable coherent sheaves on a K3 surface, when they admit a symplectic birational map of finite order with a non—trivial action on its discriminant group. Passing via Bridgeland stability, one can show these hyperkähler manifolds are itself moduli spaces of stable objects on a (possible different) K3 surface. In the second part of this talk, we deduce properties regarding the existence of birational involutions via wall—crossing and the birational geometry of these moduli spaces. This is a work in progress with Yajnaseni Dutta and Dominique Mattei.
Shizhuo Zhang, Conics on Gushel-Mukai varieties, Bridgeland moduli spaces and Lagrangian covering family
Let X be a Gushel-Mukai fourfold and Y be its general hyperplane section, which is a smooth Gushel-Mukai threefold. I realize the minimal model of Fano surface of conics on Y and the base of MRC-fibration of Hilbert scheme of conics on X as Bridgeland moduli spaces of stable objects in the Kuznetsov component of Y and X respectively. As an application, I show that the Kuznetsov component of general GM varieties determines their birational isomorphism class. Then I will outline a systematic way to produce Lagrangian subvariety of a hyperKahler variety if it is constructed as moduli space of stable objects in Kuznetsov components of cubic or Gushel-Mukai fourfolds. As an application, I will produce Lagrangian covering family for double EPW cube, which is six-dimensional HyperKahler variety. The talk is base on a series work joint with Jacovskis Augustinas, Zhiyu Liu, Hanfei Guo and Soheyla Feyzbakhsh.