Global generation of test ideals in mixed characteristic and applications (Q6609247)

Multiplier ideals (in characteristic zero) and test ideals (in positive characteristic) are well-understood and can be used to get effective global generation results in the direction of the Fujita conjecture. In mixed characteristic, the situation is less clear due to problems with localization. In this paper, the authors define a new kind of test ideal, called \(\tau_+(\mathcal O_X, \Delta)\), such that a certain global generation statement holds by definition. This, however, requires \(X\) to be projective (over a complete local ring \(R\)). In Theorem~A, they show that if \(X\) is affine, the definition is independent of the choice of compactification, and also compatible with restriction to smaller affine subsets. Therefore, \(\tau_+(\mathcal O_X, \Delta)\) can be defined for any quasi-projective scheme over \(R\).\N\NIn Theorem~B, they obtain an effective global generation result for \(\tau_+(\mathcal O_X, \Delta)\) analogous to the characteristic zero case. There are further results (Theorems~C and~D) about the asymptotic order of vanishing of linear series and about the diminished base locus of big divisors.