EQUIVARIANT RESOLUTION OF POINTS OF INDETERMINACY

Z. REICHSTEIN AND B. YOUSSIN

Abstract. We prove an equivariant form of Hironaka’s theorem on elimination of points of indeterminacy. Our argument uses canonical resolution of singularities and an extended version of Sumihiro’s equivariant Chow Lemma.

1. Introduction

Throughout this note we shall work over an algebraically closed field of characteristic 0. All algebraic varieties, schemes, groups, and all maps between them will be defined over k.

Let G be a linear algebraic group (not necessarily connected). The main objects of interest for us will be algebraic varieties with a G-action; we will refer to them as G-varieties. A G-equivariant morphism between two such varieties will be called a morphism of G-varieties. The terms ”rational map of G-varieties”, “birational morphism of G-varieties”, ”birational isomorphism of G-varieties”, etc., are defined in a similar manner.

Hironaka’s theorem on elimination of points of indeterminacy (see [Hi, §0.5, Question E and Main Theorem II]) asserts that every rational map f: X −→ Y can be resolved into a regular map by a sequence of blowups π: X_{m} ⟶ … ⟶ X_{0} = X with smooth centers. In other words, π can be chosen so that the composition fπ is regular. The purpose of this paper is to prove the following equivariant version of this result.

Theorem 1. Let f: X −→ Y be a rational map of G-varieties where Y is complete. Then there is a sequence of blowups

π: X_{m} ⟶ X_{m−1} ⟶ … ⟶ X_{1} ⟶ X_{0} = X

with smooth G-invariant centers such that the composition fπ is regular.

Our proof will rely on canonical resolution of singularities. Along the way we prove an equivariant form of Chow’s lemma (Proposition 2), generalizing a theorem of Sumihiro ([Su, Theorem 2]). For applications of Theorem 1 see [RY_{1}, Remark 5.4] and [RY_{2}, Remark 6.3].

1991 Mathematics Subject Classification. 14E15, 14L30.

Key words and phrases. group action, blowing up, resolution of singularities, equivariant map.

Z. Reichstein was partially supported by NSF grant DMS-9801675.

2. Equivariant Chow Lemma

In this section we will prove the following generalization of Chow’s lemma.

Proposition 2. For every G-variety X, there exists a quasiprojective Gvariety Z and a proper birational morphism Z ⟶ X. If X is complete then Z is projective.

Note that if G is assumed to be connected, this result is a well-known theorem of Sumihiro [Su, Theorem 2]; see also [PV, Theorem 1.3]. The argument below reduces the general case to the case where G is connected.

Proof. The second assertion is an immediate consequence of the first: if X is complete, Z ⟶ X is proper and Z is quasiprojective then Z is also complete and, hence, projective.

To prove the first assertion let G_{0} be the connected component of G. Applying Sumihiro’s theorem to X, viewed as a G_{0}-variety, yields a quasiprojective G_{0}-variety Y and a proper G_{0}-equivariant birational morphism f: Y ⟶ X.

For the rest of this proof we shall use set-theoretic notation: by a “point” we will always mean a closed point.

Recall that the homogeneous fiber product G∗_{G_{0}}Y is the G-variety defined as the geometric quotient (G×Y)/G_{0} for the action of G_{0} given by g_{0}(g, y) = (gg_{0}^{−1}, g_{0}y), where g ∈ G, g_{0} ∈ G_{0} and y ∈ Y; see [PV, Section 4.8]. We shall write [g, y] for the element of G∗_{G_{0}} Y represented by (g, y) ∈ G×Y. Since G_{0} has finite index in G, G∗_{G_{0}}Y admits a more concrete description as a disjoint union of |G/G_{0}| copies of Y. More precisely, if we choose a representative a_{h} for each h ∈ G/G_{0}, we can explicitly identify G/G_{0} × Y and G ∗_{G_{0}} Y as abstract varieties, via (h, y) ↦ [a_{h}, y]. Moreover, if we define a G-action on G/G_{0} × Y by g(h, y) ⟶ (\overline{g}h, (a_{h}^{−1}g^{−1}a_{\overline{g}h})y) then (h, y) ↦ [a_{h}, y] identifies G/G_{0} × Y and G ∗_{G_{0}} Y as G-varieties. Here \overline{g} is the image of g in G/G_{0} and (a_{h}^{−1}g^{−1}a_{\overline{g}h})y is well-defined because a_{h}^{−1}g^{−1}a_{\overline{g}h} is an element of G_{0}.

Let α: G ∗_{G_{0}} Y ⟶ X and β: G ∗_{G_{0}} Y ⟶ G/G_{0} be the maps of Gvarieties given by α: [g, y] ⟶ gf(y) and β: [g, y] ↦ \overline{g}. (Here G acts on G/G_{0} by left multiplication.) These maps are shown in the diagram below.

G ∗_{G_{0}} Y ≃ G/G_{0} × Y
α ↙  β ↘↖ s
X  G/G_{0}

Let S be the set of all sections s of β. Note that if we identify G ∗_{G_{0}} Y with G/G_{0} × Y as above, then β: G/G_{0} × Y ⟶ G/G_{0} is the projection to the first factor. Thus S ≃ Y^{|G/G_{0}|} as an abstract variety. Moreover, since β is G-equivariant, G acts on this variety by g: s ↦ t, where s, t ∈ S and t(h) = g · s(\overline{g}^{−1}h) for any h ∈ G/G_{0}.

Let Z be the closed G-invariant subvariety of S consisting of those sections s: G/G_{0} ⟶ G ∗_{G_{0}} Y ≃ G/G_{0} × Y with the property that α ∘ s(G/G_{0}) is a single point of X; we shall denote this point by x_{s}.

We claim that the morphism ϕ: Z ⟶ X given by s ⟶ x_{s} has the properties asserted in the proposition. Indeed, since Y is quasiprojective, and Z is a closed subvariety of S ≃ Y^{|G/G_{0}|}, Z is quasiprojective as well.

To show that ϕ is a birational morphism, assume the birational morphism f: Y ⟶ X is an isomorphism over a dense open subset U ⊂ X. Then V = ⋂_{h∈G/G_{0}} a_{h}U is also a dense open subset of X, and for every x ∈ V, α^{−1}(x) = {[a_{h}, f^{−1}(a_{h^{−1}}x)] : h ∈ G/G_{0}}; it is the image of the unique section s_{x} ∈ S satisfying x_{s_{x}} = x. This section is given by s_{x}(h) = [a_{h}, f^{−1}(a_{h}^{−1}x)], and the morphism V ⟶ Z, x ↦ s_{x}, is a two-sided rational inverse to ϕ.

3. Birational morphisms as blowups

Our proof of Theorem 1 in the next section will rely on the following equivariant analogue of [Ha, Theorem 7.17].

Proposition 3. Let f: X′ ⟶ X be a birational proper morphism of Gvarieties, where X is smooth and X′ is quasiprojective. Then there exists a G-invariant sheaf of ideals I on X such that X′ is the blowup of I.

Proof. Let σ: G×X ⟶ X be the given action of G on X and pr_{2} : G×X ⟶ X be the projection onto the second factor.

By a theorem of Kambayashi [Ka], there is an action of G on the projective space ℙ^{n} (via a representation G ⟶ PGL_{n+1}) and a G-equivariant embedding X′ ↪ ℙ^{n}; this yields a G-equivariant embedding i: X′ ↪ ℙ^{n}×X.

Here ℙ^{n} × X is a projective space over X; set L = i^{∗}O_{ℙ^{n}×X}(1) and T = ⨁_{d=0}^{∞} f_{∗}(L^{d}), where L^{0} = O_{ℙ^{n}×X}. Let T_{1} = f_{∗}L be the component of T of degree one. The action of G on ℙ^{n} × X yields a G-linearization of the sheaf T_{1}, i.e., an isomorphism σ^{∗}T_{1} =^{∼} pr_{2}^{∗} T_{1} which satisfies the same cocycle condition as in the definition of G-linearization of an invertible sheaf (see, e.g., [MFK, Definition 1.6]); informally speaking, G acts on the pair (X, T_{1}).

We refer to the proof of [Ha, Theorem 7.17] for the following facts:

1. After replacing the embedding i by its e-fold embedding for some positive integer e (thus replacing L by L^{e}), we may assume that the graded O_{X}-algebra T is generated by T_{1}.

2. X′ =^{∼} Proj T .

3. Assume T is generated by T_{1} as in (1). If there is an invertible sheaf M on X and a sheaf of ideals I on X such that I =^{∼} T_{1} ⊗ M, then X′ is isomorphic to the blowup of I.

The variety X is smooth, and hence, for any sheaf of ideals F on X of rank one without torsion, its dual F^{∗} = Hom(F, O_{X}) is an invertible sheaf. (To see this, note that locally at any point x ∈ X, the generator of F_{x}^{∗} is given by the homomorphism F_{x} ⟶ O_{X,x} which maps the generators of F_{x} as an O_{X,x}-module of rank one, into elements of O_{X,x} not having a nontrivial common multiple; such homomorphism is unique up to an invertible multiple since the local ring O_{X,x} is regular, and hence, factorial.)

Thus the second dual T_{1}^{∗∗} is an invertible sheaf, and we have an embedding T_{1} ↪ T_{1}^{∗∗}. The G-linearization of T_{1} yields a G-linearization of T_{1}^{∗∗}, and the above embedding is, in fact, an embedding of G-linearized sheaves. Taking M = (T_{1}^{∗∗})^{−1}, we see that X′ is isomorphic to the blowup of the sheaf of ideals I = T_{1} ⊗ (T_{1}^{∗∗})^{−1}.

The G-linearizations of T_{1} and T_{1}^{∗∗} yield a G-linearization of I and a G-linearized embedding I = T_{1} ⊗ (T_{1}^{∗∗})^{−1} ↪ T_{1}^{∗∗} ⊗ (T_{1}^{∗∗})^{−1} = O_{X}. This shows that I is a G-invariant sheaf of ideals on X.

4. Proof of Theorem 1

First of all, we may assume without loss of generality that X is smooth. Indeed, let

X_{l} ⟶^{π_{l}} … ⟶^{π_{1}} X_{0} = X ,  (1)

be the canonical resolution of singularities of X, as in [V, Theorem 7.6.1] or [BM, Theorem 13.2]. Here X_{l} is smooth, the centers C_{i} ⊂ X_{i} are smooth and G-invariant, and the action of G lifts to the entire resolution sequence (1). Replacing X by X_{l}, we may assume that X is smooth.

Now let Γ be the closure of the graph of f: X −→ Y in X × Y. By Proposition 2 there exists a proper birational morphism ϕ: Z ⟶ Γ of Gvarieties, where Z is quasi-projective. Thus we have the following diagram of G-equivariant maps:

Z ↓ ϕ Γ

↙^{α}  ↘^{β}
X  −→^{f}  Y ,

where the maps α: Γ ⟶ X and β: Γ ⟶ Y are the natural projections to the first and the second factors. Since Y is complete, α is a proper birational morphism, and consequently so is αϕ: Z ⟶ X. Proposition 3 now says that αϕ is the blow up of a G-invariant ideal sheaf I on X. By a theorem of Hironaka [Hi, Main Theorem II] there is a sequence

π: X_{m} ⟶^{π_{n}} X_{n−1} ⋯ ⟶^{π_{2}} X_{1} ⟶^{π_{1}} X_{0} = X

of blowups with smooth centers such that π^{∗}I is locally principal. Moreover, by a theorem of Bierstone and Milman [BM, Theorem 1.10] the sequence π can be chosen in a canonical way; in particular, the center of each π_{i} can be chosen to be smooth and G-invariant; see [BM, Remark 1.5]. Now by the universal property of blowing up (see [Ha, Proposition II.7.14]), the composition (αϕ)^{−1}π: X_{m} −→ Z is a regular map. Hence, fπ = βϕ(αϕ)^{−1}π: X_{m} −→ Y is also regular, as claimed.

Acknowledgements

Some of the work on this paper was done while the first author was participating in the program on Algebraic Groups, Invariant Theory and Applications at the Erwin Schrödinger Institute in Vienna, Austria in the Fall of 2000. He would like to thank the organizers of this program and the Institute staff for their hospitality and support. The second author warmly thanks Institute of Mathematics of Hebrew University for its hospitality during 1999/2000. We would also like to thank an anonymous reviewer whose comments on an earlier version of this paper allowed us to simplify the proof of Theorem 1.

References

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∗ Available at http://ucs.orst.edu/˜ reichstz/pub.html.

Department of Mathematics, Oregon State University Corvallis, OR 97331 Current address: 1498 Meadows Drive, Lake Oswego, OR 97034, USA

E-mail address: zinovy@math.orst.edu

Department of Mathematics and Computer Science, University of the Negev, Be’er Sheva’, Israel

Current address: Hashofar 26/3, Ma’ale Adumim, Israel

E-mail address: youssin@math.bgu.ac.il