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Quotients of schemes by $$\alpha _p$$ or $$\mu _{p}$$ actions in characteristic $$p>0$$. (English) Zbl 1386.14175
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:
(i)
the existence of the quotient $$\pi: X \to Y$$ and inseparability of $$\pi$$ by D. Mumford [Abelian varieties. London: Oxford University Press (1970; Zbl 0223.14022)] and by T. Ekedahl [Publ. Math., Inst. Hautes Étud. Sci. 67, 97–144 (1988; Zbl 0674.14028)];
(ii)
vector fields and inseparable morphisms of algebraic surfaces by A. N. Rudakov and I. R. Shafarevich [Izv. Akad. Nauk SSSR, Ser. Mat. 40, 1269–1307 (1976; Zbl 0365.14008)];
(iii)
canonically polarized varieties and schemes by T. Matsusaka [Am. J. Math. 92, 283–292 (1970; Zbl 0195.22802)] and by others;
(iv)
non-smooth Picard scheme by J.-i. Igusa [Proc. Natl. Acad. Sci. USA 41, 964–967 (1955; Zbl 0067.39102)], by M. Raynaud [Astérisque 64, 87–148 (1979; Zbl 0434.14024)], by C. Liedtke [Math. Z. 259, No. 4, 775–797 (2008; Zbl 1157.14023)] and by others.
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:
(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 $$Y$$;
(b)
next he demonstrates that $$Y$$ has Gorenstein singularities in codimension 1;
(c)
from that and from Serre’s condition $$S_2$$ for $$X$$ he obtains that $$\omega_X = {\pi}^{*}\cdot \omega_Y \otimes {\pi}^{!} {\mathcal O}_Y$$;
(d)
next he proves that $$({\pi}^{!} {\mathcal O}_Y)^* = I_{\mathrm{fix}}^{[p-1]}$$ and concludes the proof of the theorem.
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 $$p>2$$.

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 14L15 Group schemes 14J15 Moduli, classification: analytic theory; relations with modular forms 14J50 Automorphisms of surfaces and higher-dimensional varieties
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##### References:
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