Conference Global Portuguese Mathematicians 

13/07/2017, 17:30 — 18:30 — Abreu Faro Amphitheatre
Inês Henriques, University of Sheffield

Test, Invariant and multiplier ideals

As a measure of singularities of a scheme $X$, we study the log-canonical threshold, defined over the complex numbers in terms of integrability, or over any field of characteristic $0$ in terms of resolution of singularities.

In positive characteristic, one can define the $F$-pure threshold in terms of the Frobenius endomorphism. Quite surprisingly, these two different ways of quantifying singularities are intimately related: If the equations defining $X$ have integral coefficients, then the $F$-pure threshold of the reduction of $X$ modulo a prime number $p$ approaches the $\log$-canonical threshold of $X$ as $p$ goes to infinity.

We will discuss basic properties of these invariants in terms of the corresponding multiplier and test ideals, and present work done with Matteo Varbaro, on the multiplier ideals of $G$-stable projective subschemes of $\operatorname{Proj}(\operatorname{Sym}(E))$, when $E$ is the tensor product between $2$ finite vector spaces $V$ and $W$ over a field of characteristic $0$, and $G = \operatorname{GL}(V) \times \operatorname{GL}(W)$. The proof, quite surprisingly relies on a reduction to the context of positive characteristic, where $G$-stable subschemes are not well-understood!