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arXiv · 2501.13613

The defect of the F-pure threshold

Alessandro De Stefani·Luis Núñez-Betancourt·Ilya Smirnov
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Abstract

Introduced by Takagi and Watanabe, the F-pure threshold is an invariant defined in terms of the Frobenius homomorphism. While it finds applications in various settings, it is 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 $(R,\mathfrak{m})$ by setting ${\rm dfpt}(R)=\dim (R) - {\rm fpt}(\mathfrak{m})$. 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.

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Alessandro De Stefani, Luis Núñez-Betancourt, Ilya Smirnov. 2025-01-23. The defect of the F-pure threshold. https://arxiv.org/abs/2501.13613

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