The defect of the F-pure threshold

Introduced by Takagi and Watanabe, F-pure thresholds are invariants defined in terms of the Frobenius homomorphism. While they find applications in various settings, they are primarily used as a local invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the defect of the F-pure threshold of a local ring (R,m) by setting dfpt(R)=dim⁡(R)−fpt(m). It turns out that this invariant defines an upper semi-continuous function on a scheme and satisfies Bertini-type theorems. We also study the behavior of the defect of the F-pure threshold under flat extensions and after blowing up the maximal ideal of a local ring.