Universal flattening of Frobenius (Q2881356)

Let \(X\) be a \(d\)-dimensional variety of positive characteristic \(p >0\) over a perfect field \(k\). The author defines the \(e\)th \(F\)-blowup of \(X\), \(FB_e(X)\), and shows that there exists a natural projective morphism \(\pi_e: FB_e(X) \to X\) which is an isomorphism exactly over the smooth locus of \(X\), \(X_{sm}\). The author develops the basic properties of the \(F\)-blowup, and formulates a number of natural questions for the sequence of \(F\)-blowups obtained. He shows that, in dimension one, \(FB_e(X)\) is smooth for all \(e \gg 0\). The toric case is investigated in detail, over arbitrary fields. A new variety \(FB_{(l)}(X)\) is defined that is toric and such that, if \(k\) is perfect, \(FB_{(l)}(X) =FB_e(X)\) for \(l =p^e\). Among other things, it is shown that the sequence of \(F\)-blowups of a toric variety is bounded, and, for a normal two-dimensional toric variety, \(FB_{(l)}(X)\) is the minimal resolution of \(X\) for \(l \gg 0\). Connections to the \(G\)-Hilbert scheme defined by \textit{Y. Ito} and \textit{I. Nakamura} [Proc. Japan Acad., Ser. A 72, No. 7, 135--138 (1996; Zbl 0881.14002)] are also explored. It is proved that the \(G\)-Hilbert scheme of a quotient variety \(X=M/G\) by a finite group \(G\) such that \(p\) does not divide \(|G |\) is isomorphic to \(FB_e(X)\) for sufficiently large \(e\).