Bertini theorems for F-singularities

We prove that strongly F-regular and F-pure singularities satisfy Bertini-type theorems (including in the context of pairs) by building upon a framework of Cumino, Greco and Manaresi (compare with the work of Jouanolou and Spreafico). We also prove that F-injective singularities fail to satisfy even the most basic Bertini-type results.Comment: Typos corrected and other minor changes. To appear in the Proceedings of the London Mathematical Societ

Spontaneous emission of polaritons from a Bose-Einstein condensate

We study the spontaneous emission of a partially excited Bose-Einstein condensate composed of two-level atoms. The formation of polaritons induced by the ground-state part of the condensate leads to an avoided crossing in the photon spectrum. This avoided crossing acts similarly to a photonic band gap and modifies the spontaneous emission rate.Comment: 4 pages, 2 figures, revte

F-singularities in families

We study the behavior of test ideals and F-singularities in families. In particular, we obtain generic (and non-generic) restriction theorems for test ideals and non-F-pure ideals. Additionally, we study the global behavior of certain canonical linear systems (induced by Frobenius) associated to adjoint line bundles, in families. As a consequence, we obtain some positivity results for pushforwards of some adjoint line bundles and for certain subsheaves of these.Comment: 60 pages, typos corrected throughout and improved exposition, to appear in Algebraic Geometr

Discreteness and rationality of FF-jumping numbers on singular varieties

We prove that the $F$-jumping numbers of the test ideal \tau(X; \Delta, \ba^t) are discrete and rational under the assumptions that $X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+\Delta$ is \bQ-Cartier of index not divisible $p$, and either $X$ is essentially of finite type over a field or the sheaf of ideals \ba is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.Comment: 29 pages, minor changes, to appear in Mathematische Annale