F-blowups of F-regular surface singularities (Q2845037)

Hereafter, we refer to [\textit{R. Hartshorne}, Algebraic geometry. York-Heidelberg-Berlin: Springer-Verlag (1977; Zbl 0367.14001)] for any unexplained terminology.NEWLINENEWLINELet \(\pi: Y\rightarrow X\) be a morphism of reduced schemes of finite type over a field \(\mathbb{K}\) and suppose that \(\mathfrak{a}\) is a sheaf of ideals on \(X\). We say that \(\pi\) is a log resolution of \(\mathfrak{a}\) provided it satisfies the following four conditions. {\parindent=6mm \begin{itemize}\item[1.] \(\pi\) is birational and proper. \item[2.] \(Y\) is smooth over \(\mathbb{K}\). \item[3.] \(\mathfrak{a}\mathcal{O}_Y =\mathcal{O}_Y (-G)\) is an invertible sheaf corresponding to a divisor, namely \(-G\). \item[4.] If \(E\) is the exceptional set of \(\pi\), then \(\operatorname{Supp} (G)\cup E\) has simple normal crossings. NEWLINENEWLINE\end{itemize}} Moreover, we say that \(\pi\) is a strong log resolution if it is a log resolution and \(\pi\) is an isomorphism outside of the subscheme \(V(\mathfrak{a})\) defined by \(\mathfrak{a}\). It is well known, by the celebrated results obtained by \textit{H. Hironaka} in [Ann. Math. (2) 79, 109--203, 205--326 (1964; Zbl 0122.38603)], that log resolutions always exist if the characteristic of \(\mathbb{K}\) is zero and strong log resolutions always exist in case \(X\) is smooth and \(\mathbb{K}\) has characteristic zero. However, despite the effort of many researchers the existence of resolution of singularities over a field of prime characteristic is still an open question in general.NEWLINENEWLINELoosely speaking, in characteristic zero Hironaka showed that it is possible to obtain a (not necessarily unique) resolution of singularities through a sequence of normalizations and blowups at smooth centers. It turns out that this strategy does not work in prime characteristic; regardless, it is natural to ask the following:NEWLINENEWLINEQuestion. Is there a notion in prime characteristic which might play the role of blowup in characteristic zero?NEWLINENEWLINEInspired by this question, \textit{T. Yasuda} in [Am. J. Math. 134, No. 2, 349--378 (2012; Zbl 1251.14002)] defined the so-called \(e\)th \(F\)-blowup, with the hope that it might play the same role as the blowup in characteristic zero. Roughly speaking, the idea is, given a (possibly singular) algebraic variety over a field of prime characteristic, to construct another algebraic variety \(Y\) on which the Frobenius endomorphism is flat. The idea is, of course, inspired by the classical result obtained by \textit{E. Kunz} in [Am. J. Math. 91, 772--784 (1969; Zbl 0188.33702)] which characterizes the regularity of a commutative reduced ring \(R\) of prime characteristic in terms of the fact that the Frobenius map on \(R\) is flat.NEWLINENEWLINEYasuda's definition works as follows. Let \(\mathbb{K}\) be an algebraically closed field of prime characteristic, and let \(X\) be an algebraic variety over \(\mathbb{K}\) of dimension \(n\). The \(e\)th \(F\)-blowup \(\operatorname{FB}_e (X)\) of \(X\) is defined to be the closure of the subset NEWLINE\[NEWLINE \{ (F^e)^{-1} (x)\mid x\in X\text{ smooth}\}\subseteq\operatorname{Hilb}_{p^{ne}} (X^{(e)}), NEWLINE\]NEWLINE where \(X^{(e)}\) is the ringed space \((X,F_*^e \mathcal{O}_X)\) and \(\operatorname{Hilb}_{p^{ne}} (X^{(e)})\) denotes the Hilbert scheme of zero dimensional subschemes of \(X^{(e)}\) of length \(p^{ne}\). We refer to [Zbl 0188.33702] for more details.NEWLINENEWLINEFrom now on, we restrict our attention to the case of surfaces. In this case, given a surface \(S\) it is known that, in any characteristic, there exists a minimal resolution of singularities \(S_0\rightarrow S\). So, it is natural to ask the following:NEWLINENEWLINEQuestion. Let \(\mathbb{K}\) be an algebraically closed field of prime characteristic, let \((S,x)\) be a normal surface singularity and let \(S_0\rightarrow S\) be the minimal resolution. When is \(\operatorname{FB}_e (S)\) equal to the minimal resolution \(S_0\)?NEWLINENEWLINEIt is true that \(\operatorname{FB}_e (S)=S_0\) for \(e\gg 0\) if either \(S\) is a toric singularity, a tame quotient singularity, or an \(F\)-regular double point.NEWLINENEWLINEWe wish to introduce an additional notion before starting properly our review. Let \(R\) be an integral domain of prime characteristic \(p\) which is \(F\)-finite. We say that \(R\) is strongly \(F\)-regular if, for any nonzero element \(c\in R\), there exists a power \(q=p^e\) such that the inclusion map \(c^{1/q}R\hookrightarrow R^{1/q}\) splits as an \(R\)-module homomorphism.NEWLINENEWLINEIn the paper under review, the author shows as main result that if \((S,x)\) is an strongly \(F\)-regular surface singularity, then its \(e\)th \(F\)-blowup \(\operatorname{FB}_e (S)\) coincides with the minimal resolution of \(X\) for \(e\gg 0\). This result generalizes an earlier one obtained by \textit{N. Hara} and \textit{T. Sawada} in [RIMS Kôkyûroku Bessatsu B24, 121--141 (2011; Zbl 1228.13009)] which was proved for \(F\)-rational double points. One of the technical tools developed by the author for that purpose, which is interesting in its own right, is a nice characterization of complete strongly \(F\)-regular rings of dimension \(2\) over \(\mathbb{K}\) (cf. Theorem 2.1). Finally, the author raises several questions related with these topics in order to stimulate further research about \(F\)-blowups and strongly \(F\)-regular rings.