Now showing items 1-10 of 35
Cubic fourfolds containing a plane and a quantic del Pezzo surface
We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class β of the even ...
The Picard group of the moduli space of curves with level structures
For 4 - L and g large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level L structures. In particular, we determine the divisibility properties of ...
Higher-Order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders
(Oxford University Press, 2012)
The second rational homology group of the moduli space of curves with level structures
Let Γ be a finite-index subgroup of the mapping class group of a closed genus g surface that contains the Torelli group. For instance, Γ can be the level L subgroup or the spin mapping class group. We show that H2(Γ;Q) ∼= Q ...
On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data
We consider the KdV equation ∂tu+∂3xu+u∂xu=0 with quasi-periodic initial data whose Fourier coefficients decay exponentially and prove existence and uniqueness, in the class of functions which have an expansion with ...
Abelian quotients of subgroups of the mapping class group and higher Prym representations
A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto Z if the genus of the surface is large. We prove that if this conjecture holds ...
A Birman exact sequence for Aut(Fn)
The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free ...
Small generating sets for the Torelli group
Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup Ig of the genus g mapping class group has a finite generating set whose size grows cubically with respect to g. Our main tool is a new space called ...
Failure of the Hasse Principle on General K3 Surfaces
We show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general K3 surface X of degree 2 over Q, together with a two-torsion Brauer class that is ...