Main | Venue | Participants | Program | Abstracts | Getting Here |
I will give an overview of the geometry of Fano 3-folds after Mori theory. Building on recent advances in explicit geometry of Fanos, I introduce a new viewpoint on the classification of terminal Fano 3-folds. This will be exhibited by explicit examples.
It is conjectured that the space of stability conditions of a triangulated category may be endowed with a Frobenius manifold structure. We will view how to construct a weaker structure, that is a formal Frobenius type structure, starting from the stability data itself and the Joyce's generating function for Donaldson-Thomas type invariants.
The crepant resolution conjecture is a conjecture in enumerative geometry originating from string theory. It relates the Donaldson-Thomas invariants of a three-dimensional Calabi-Yau orbifold to those of a particular crepant resolution of its coarse moduli space. In this talk, we will explain the statement of this conjecture and look at a number of techniques that could help one prove it.
I'll describe joint work with Yukari Ito and Joseph Karmazyn where we provide the natural moduli space description of the minimal resolution \(Y\) of a surface singularity \(A^2/G\) for any finite subgroup \(G\) in \(\mathrm{GL}(2)\). In fact, our results apply to any subminimal partial resolution. The approach blends geometry and algebra by constructing every such space as the multigraded linear series of a vector bundle on \(Y\). Time permitting, I'll explain how Reid's recipe helps suggest efficient moduli space descriptions in dimension three; or we might just decide to go to the pub instead.
We show how the infinitesimal Abel-Jacobi map can be realised as a morphism of formal deformation functors (and so as a morphism in the homotopy category of differential graded Lie algebras). The whole construction is carried out in a general setting, of which the classical Abel-Jacobi map is a special example. Joint work with Marco Manetti.
TBA
Under mirror symmetry, Fano varieties up to deformation correspond to certain Laurent polynomials up to mutation. If proven, this correspondence could be used in the classification of Fano varieties: this is part of a project involving Coates, Corti, Galkin, Golyshev, Kasprzyk, Tveiten, and others. The correspondence and the classification have been completed in dimension up to 3. In this talk, I will discuss joint work with Coates on dimension 4: why quiver flag varieties are a natural and combinatorially attractive type of ambient space to consider, and the 138 new Fano fourfolds we have found so far.
In a joint work with Klaus Hulek, we study the set \(R_g\) of possible Picard numbers of abelian varieties of dimension \(g\). By the Lefschetz Theorem on (1,1)-classes, \(R_g \subseteq [1,g^2]\cap \mathbb N\). We prove non-completeness of Picard numbers, namely that for every \(g \geq 3\), this inclusion is strict. We also study the asymptotic structure of \(R_g\) as \(g \rightarrow +\infty\) by proving an asymptotic completeness result for Picard numbers, and by describing the distribution of large Picard numbers in \([1,g^2]\) as \(g\) grows. As a byproduct, we deduce a structure theorem for abelian varieties with large Picard number.
I will present the construction of some special crepant resolutions of the quotient of the four-dimensional affine space \(\mathbb{C}^4\) by the group \(G = ( \mathbb{Z}/r )^3\) and show how these resolutions relate to the Hilbert scheme of \(G\)-orbits \(G\)-Hilb.
Let \(X\) be a Mori Dream Space embedded in a toric variety with algebraic torus \(T\). We construct a tropical compactification of the restriction of \(X\) to \(T\) that determines a model of \(X\) dominating all its small \(\mathbb{Q}\)-factorial modifications. Exploiting the combinatorial properties of such compactification, we recover a Minkowski basis for the Newton-Okounkov bodies of Cartier divisors on \(X\) and hence a set of generators of the movable cone of \(X\). This is a joint work with Stefano Urbinati.
One way to construct minimal surfaces of general type are as abelian Galois covers of other (possibly singular) surfaces, ramified along a collection of divisors. By studying the deformation theory of theses covers, we gained a deeper insight into the structure of the moduli spaces of surfaces of general type. We expect the equivariant derived category of these covering surfaces to have a semi-orthogonal decomposition with pieces given by the minimal resolution of the base surface and the branch divisors. We will give examples of such semi-orthogonal decompositions and describe how they are constructed using the language of stacks.
Tautological classes are geometrically defined classes in the Chow ring of the moduli space of curves which are particularly well understood. The classes of many known geometrically defined loci were proven to be tautological. A bielliptic curve is a curve with a 2-to-1 map to an elliptic curve. In this talk we will build on an idea of Graber and Pandharipande to show that the closure of the locus of bielliptic curves in the moduli space of stable curves of genus \(g\) is non-tautological when \(g\) is at least 12.