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Wonderful compactification of an arrangement of subvarieties. (English) Zbl 1187.14060

The author defines the so-called wonderful compactification of an arrangement of subvarieties. Let \(Y\) be a smooth algebraic variety (over an arbitrary algebraically closed field); an arrangement of subvarieties is a finite collection of smooth subvarieties which is closed under the scheme-theoretic intersection. One can also define building sets; in particular, it is a collection of subvarieties such that the set of all intersection of subvarieties in it is an arrangement of subvarieties.
The author associates to each (nonempty) building set \(\mathcal{G}\) of \(Y\) a smooth variety birational to \(Y\), the so-called wonderful compactification \(Y_{\mathcal{G}}\) of \(\mathcal{G}\). It is a compactification of the open subvariety \(Y^{0}:=Y\setminus\bigcup_{G\in\mathcal{G}}G\) of \(Y\). The complementary set \(Y_{\mathcal{G}}\setminus\, Y^{0}\) is a union of nonsingular prime divisors indexed by \(\mathcal{G}\) and such that any set of such divisors intersects transversally. The author describes also when such an intersection is non-empty (see Theorem 1.2). Such properties are proved using a construction of \(Y_{\mathcal{G}}\) through an explicit sequence of blow-ups along smooth centers.
Roughly speaking these centers correspond to the elements of \(\mathcal{G}\). The author proves that the blow-up of \(Y\) along a minimal element of \(\mathcal{G} \) has a natural building set whose elements are the dominant transforms of the subvarieties in \(\mathcal{G}\) (see Proposition 2.8). Then, he uses this fact to construct recursively \(Y_{\mathcal{G}}\). In Theorem 1.3 the author explains how to alter the order of blow-ups obtaining the same variety. In Proposition 5.3, the dimensions of the centers of the blow-ups are determined.
The wonderful compactification of this paper generalizes the following constructions:
- the Fulton-Macpherson configuration space [see W. Fulton, R. MacPherson, Ann. Math. (2) 139, No.1, 183–225 (1994; Zbl 0820.14037)];
- De Concini-Procesi’s wonderful model of subspace arrangements (see the above-mentioned [C. De Concini, C. Procesi, Sel. Math., New Ser. 1, No. 3, 459–494 (1995; Zbl 0842.14038)];
- Ulyanov’s polydiagonal compactification [see A. P. Ulyanov, J. Algebr. Geom. 11, No. 1, 129–159 (2002; Zbl 1050.14051)];
- the Kuperberg-Thurston’s compactification \(X^{\Gamma}\) [see G. Kuperberg, D. Thurston, Pertubative 3-manifold invariants by cut-and-paste topology, preprint, arXiv:math.GT/9912167];
- the moduli space \(\overline{M}_{0,n}\) of rational curves with \(n\) marked points [see S. Keel, Trans. Am. Math. Soc. 330, No.2, 545–574 (1992; Zbl 0768.14002)] for a description of this space as a wonderful compactification);
- Hu’s compactification of open subvarieties [see Y. Hu, Trans. Am. Math. Soc. 355, No. 12, 4737–4753 (2003; Zbl 1083.14004)].
In the section 4, the author explains the relations between the above-mentioned constructions and how to re-obtain them using Theorems 1.2 and 1.3. For example the construction of Kuperberg-Thurston is a generalization of the one of Fulton-Macpherson. Moreover, in some case the Theorem 1.2 can be used to give some insights to the previous constructions. For example, it can be used to construct Fulton-Macpherson through a more symmetric sequence of blow-ups. Another example is the following: the varieties of Kuperberg-Thurston are associated to a graph with some appropriate properties. The author shows that, given two such graphs \(\Gamma_{1}\subsetneq\Gamma_{2}\), the variety associated to \(\Gamma_{2}\) is obtained from the one associated to \(\Gamma_{1}\) by a sequence of blow-ups.
It is also shown that the Hu compactification is maximal. More precisely, given any building set \(\mathcal{G}\) with the same arrangement of the one associated to the building set \(S\) used by Hu, the natural birational map \(Y_{S}\dashrightarrow Y_{\mathcal{G}}\) can be extended to a morphism. Instead, the construction of Fulton-Macpherson, Kuperberg-Thurston and \(\overline{M}_{0,n}\) are minimal.
The author acknowledges the following paper as his principal inspiration: [R. MacPherson, C. Procesi, Sel. Math., New Ser. 4, No. 1, 125–139 (1998; Zbl 0934.32014)]. But, even in the complex case, this work is neither strictly more general nor strictly less general than the one of Macpherson and Procesi (observe that, in this last work, the varieties are all complex). For example, in the paper of MacPherson and Procesi it is not assumed the smoothness of the varieties in \(\mathcal{G}\).
Remark that the proof of the more technical lemmas is postponed to the appendix.

MSC:

14N20 Configurations and arrangements of linear subspaces
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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References:

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