Upcoming Talks

The rationality problem for smooth Fano threefolds over algebraically closed fields is basically solved.
In this talk I will discuss rationality  of forms of these  Fanos over nonclosed fields of characteristic 0.
I will concentrate on the case where the  Picard number  equals 1.

The talk is based on joint work with Alexander Kuznetsov (in progress).

For any $k\geq 1$, we study the distribution of the difference between the number of integers $n\leq x$ with $\omega(n)=k$  or $\Omega(n)=k$ in two different arithmetic progressions, where $\omega(n)$ is the number of distinct prime factors of $n$ and $\Omega(n)$ is the number of prime factors of $n$ counted with multiplicity . Under some reasonable assumptions, we show that, if $k$ is odd, the integers with $\Omega(n)=k$ have preference for quadratic non-residue classes; and if $k$ is even, such integers have preference for quadratic residue classes.

There are three possibilities for the quotient of the Brauer group of X modulo constants when X is a del Pezzo surface of degree four over the rational numbers . In this talk we will explain how often each of them occurs when X ranges across a family of quartic del Pezzo surfaces equipped with a conic bundle structure. We will also give an explicit description of the generators of this quotient which allows us to calculate the frequency of such surfaces violating the Hasse principle. This talk is based on a joint work in progress with Cecília Salgado.