Quotients of schemes by \(\alpha _p\) or \(\mu _{p}\) actions in characteristic \(p>0\) (Q506997)

Let \(k\) be a field of characteristic \(p > 0\), let \(X\) be an integer scheme over \(k\) which admits a nontrivial \(\alpha_p\) or \(\mu_p\) action and let \(\pi: X \to Y\) be the quotient under this action. In the paper, which is divided into seven sections, the author study the structure of the map \(\pi\) and of the quotient \((Y,\pi)\). The main result of the paper under review is an adjunction formula (Theorem 6.1). The paper is concerned with the following problems and results: {\parindent=8mm \begin{itemize}\item[(i)] the existence of the quotient \(\pi: X \to Y\) and inseparability of \(\pi\) by \textit{D. Mumford} [Abelian varieties. London: Oxford University Press (1970; Zbl 0223.14022)] and by \textit{T. Ekedahl} [Publ. Math., Inst. Hautes Étud. Sci. 67, 97--144 (1988; Zbl 0674.14028)]; \item[(ii)] vector fields and inseparable morphisms of algebraic surfaces by \textit{A. N. Rudakov} and \textit{I. R. Shafarevich} [Izv. Akad. Nauk SSSR, Ser. Mat. 40, 1269--1307 (1976; Zbl 0365.14008)]; \item[(iii)] canonically polarized varieties and schemes by \textit{T. Matsusaka} [Am. J. Math. 92, 283--292 (1970; Zbl 0195.22802)] and by others; \item[(iv)] non-smooth Picard scheme by \textit{J.-i. Igusa} [Proc. Natl. Acad. Sci. USA 41, 964--967 (1955; Zbl 0067.39102)], by \textit{M. Raynaud} [Astérisque 64, 87--148 (1979; Zbl 0434.14024)], by \textit{C. Liedtke} [Math. Z. 259, No. 4, 775--797 (2008; Zbl 1157.14023)] and by others. \end{itemize}} Sections 1 and 2 of the paper under review are on the introduction and basic definitions. Section 3 is on existence and basic properties of the quotient, Sections 4 and 5 on quotients by \(\mu_p\) and \(\alpha_p\) actions. Section 6 deals with adjunction formulas. The author's method of proof of Theorem 6.1 is as follows: {\parindent=6mm \begin{itemize}\item[(a)] using properness of \(X\) and properties of the quotient map \(\pi: X \to Y\) he obtains the existance of a dualizing sheaf \(\omega_Y\) on \(Y\); \item[(b)] next he demonstrates that \(Y\) has Gorenstein singularities in codimension 1; \item[(c)] from that and from Serre's condition \(S_2\) for \(X\) he obtains that \(\omega_X = {\pi}^{*}\cdot \omega_Y \otimes {\pi}^{!} {\mathcal O}_Y\); \item[(d)] next he proves that \(({\pi}^{!} {\mathcal O}_Y)^* = I_{\mathrm{fix}}^{[p-1]}\) and concludes the proof of the theorem. \end{itemize}} The last section contains interesting observations on non-smooth Picard scheme of smooth canonically polarized surfaces over algebraically closed fields of characteristics \(p = 2\) and \(p>2\).