Finite torsors over strongly FF-regular singularities

In this paper, we extend the work by K. Schwede, K. Tucker, and the author on the local \'etale fundamental group of strongly $F$-regular singularities. Let $k$ be an algebraically closed field of positive characteristic. We study the existence of finite torsors over the regular locus of a strongly $F$-regular $k$-germ $(R,\mathfrak{m},k)$ that do not come from restricting a torsor over the whole spectrum. Concretely, we prove that there exists a finite cover $R \subset R^{\star}$ with the following properties: $R^{\star}$ is a strongly $F$-regular $k$-germ, and for all finite group-schemes $G/k$ with solvable connected-component-at-the-identity, every $G$-torsor over the regular locus of $R^{\star}$ extends to a $G$-torsor over the whole spectrum. To achieve this, we obtain a generalized transformation rule for the $F$-signature under finite extensions. This formula also proves that degree-$n$ Veronese-type cyclic covers over $R$ stay strongly $F$-regular with $F$-signature $n \cdot s(R)$. Similarly, this transformation rule is used to show that the torsion of the divisor class group of $R$ is bounded by $1/s(R)$. By taking cones, we show that the torsion of the divisor class group of a globally $F$-regular $k$-variety is bounded in terms of $F$-signatures.Comment: 35 pages, the solvable case was completed, comments are so much welcom

On the behavior of FF-signatures, splitting primes, and test modules under finite covers

We give a comprehensive treatment on how certain fundamental objects in Cartier theory such a $F$-signatures, splitting primes, splitting ratios, and test modules behave under finite covers. We recover previously known results as particular instances. To this end, we expand on the notion of transposability along a section section of the relative canonical module as first introduced by K.~Schwede and K.~Tucker.Comment: 49 pages, comments are more than welcome. v2: we removed normality from our hypothesis, expanded on the concept of transposability, added a new transformation rule for adjoint-like ideals. There are substantial changes from v

On the monotonicity of the correction term in Ramanujan's factorial approximation

We present two new proofs of the monotonicity of the correction term $\theta_n$ in Ramanujan's refinement of Stirling's formula.Comment: Latex, 5 page

Singularities of determinantal pure pairs

Let $X$ be a generic determinantal affine variety over a perfect field of characteristic $p \geq 0$ and $P \subset X$ be a standard prime divisor generator of $\mathrm{Cl}(X) \cong \mathbb{Z}$. We prove that the pair $(X,P)$ is purely $F$-regular if $p>0$ and so that $(X,P)$ is purely log terminal (PLT) if $p=0$ and $(X,P)$ is log $\mathbb{Q}$-Gorenstein. In general, using recent results of Z. Zhuang and S. Lyu, we show that $(X,P)$ is of PLT-type, i.e. there is a $\mathbb{Q}$-divisor $\Delta$ with coefficients in $[0,1)$ such that $(X,P+\Delta)$ is PLT.Comment: 23 pages, minor corrections, final version, accepted for publication at the Bollettino dell'Unione Matematica Italian

On the behavior of stringy motives under Galois quasi-\'etale covers

We investigate the behavior of stringy motives under Galois quasi-\'etale covers in dimensions $\leq 3$. We prove them to descend under such covers in a sense defined via their Poincar\'e realizations. Further, we show such descent to be strict in the presence of ramification. As a corollary, this reduces the problem regarding the finiteness of the \'etale fundamental group of KLT singularities to a DCC property for their stringy motives. We verify such DCC property for surfaces. As an application, we give a characteristic-free proof for the finiteness of the \'etale fundamental group of log terminal surface singularities, which was unknown at least in characteristics $2$ and $3$.Comment: 25 pages, 6 figures, comments are very much welcome, some typos were fixed, some examples were adde

Varieties with ample Frobenius-trace kernel

In the search of a projective analog of the Kunz's theorem and a Frobenius-theoretic analog of the Hartshorne--Mori's theorem, we investigate the positivity of the kernel of the Frobenius trace (equivalently, the negativity of the cokernel of the Frobenius endomorphism) on a smooth projective variety over an algebraically closed field of positive characteristic. For instance, such kernel is ample for projective spaces. Conversely, we show that for curves, surfaces, and threefolds the Frobenius trace kernel is ample only for Fano varieties of Picard rank $1$.Comment: 38 pages, comments are very much welcome

On the local \'etale fundamental group of KLT threefold singularities

Let $S$ be KLT threefold singularity over an algebraically closed field of positive characteristic $p>5$. We prove that its local \'etale fundamental group is tame and finite. Further, we show that every finite unipotent torsor over a big open of $S$ is realized as the restriction of a finite unipotent torsor over $S$.Comment: 34 pages, comments are welcome! New appendix by J\'anos Koll\'ar which removes all rationality hypothesis from our arguments on finitenes