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Ordinarity of configuration spaces and of wonderful compactifications. (English) Zbl 1238.14016

Let \(X\) be a smooth, projective variety over an algebraically closed field \(k\). A finite collection \(S\) of closed, smooth subvarieties is an arrangement if the scheme theoretic intersection of any elements of \(S\) is an element of \(S\) (if not empty). Next, one can define a building set \(G\subset S\); in particular, this subset is such that any element of \(S\) is the transversal intersection of elements of \(G\). The wonderful compactification of \((X,G)\) is the closure of \(X\setminus \bigcup_{Y\in G}Y\) in \(\prod_{Y\in G}\mathrm{Bl}_Y(X)\). It is a smooth, projective variety (see [C. De Concini and C. Procesi, Sel. Math., New Ser. 1, No. 3, 459–494 (1995; Zbl 0842.14038)] and [L. Li, Mich. Math. J. 58, No. 2, 535–563 (2009; Zbl 1187.14060); arXiv:math/0611412]).
Suppose now that \(k\) has positive characteristic. A smooth, projective variety \(X\) is said to be ordinary if all the Rham-Witt complexes \(H^i(X,BW\Omega^j_X)\) vanish. The vanishing of these is equivalent to the vanishing of the Zariski cohomology groups \(H^i(X,d\Omega_X^{j-1})\) of the sheaf of locally exact forms. Some example of ordinary varieties are projective spaces and generalized flag varieties. For abelian varieties, ordinarity in the above sense is equivalent to ordinarity in the usual sense (invertibility of the Hasse-Witt matrix); a general abelian variety is ordinary with a suitable polarization by P. Norman and F. Oort, Ann. Math. (2) 112, 413–439 (1980; Zbl 0483.14010)]. A general complete intersection in projective space is ordinary by [L. Illusie, in: The Grothendieck Festschrift, Vol. II, Prog. Math. 87, 375–405 (1990; Zbl 0728.14021)].
A building set is ordinary if all the scheme theoretic intersections of any members of \(G\) are ordinary (when not-empty).
The author proves that \(X_G\) is ordinary if and only if \(G\) is ordinary.
In particular the following schemes are ordinary: the configuration space of [W. Fulton and R. MacPherson, Ann. Math. (2) 139, No. 1, 183–225 (1994; Zbl 0820.14037)] and its generalizations [B. Kim and F. Sato, Sel. Math., New Ser. 15, No. 3, 435–443 (2009; Zbl 1177.14029)]; the moduli space of \(n\)-pointed curves of genus zero [S. Keel, Trans. Am. Math. Soc. 330, No. 2, 545–574 (1992; Zbl 0768.14002)]; the compactification of Ulyanov [A. Ulyanov, J. Algebr. Geom. 11, No. 1, 129–159 (2002; Zbl 1050.14051)]; the compactification of Kuperberg-Thurston [Zbl 1187.14060]; the space of stable pointed rooted trees of [L. Chen, A. Gibney and D. Krashen, J. Algebr. Geom. 18, No. 3, 477–509 (2009; Zbl 1171.14009)]; the compactification of open varieties of [Y. Hu, Trans. Am. Math. Soc. 355, No. 12, 4737–4753 (2003; Zbl 1083.14004)].

MSC:

14G17 Positive characteristic ground fields in algebraic geometry
14J99 Surfaces and higher-dimensional varieties
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