The goal of §4is to reduce the proof of Theorem1.1to the case when the following conditions are satisfied:
(1)Xis regular, (2) the log structure is given by an snc divisorZwhich isG-strictin the sense that for anyg∈G and a componentZieithergZi = ZiorgZi∩Zi =∅, (3)Gacts freely onX\Zand for any geometric point x→Xthe inertia groupGxis abelian.
4.1. Plan. — A general method for constructing aG-equivariant morphismfas in Theorem1.1is to construct a tower ofG-equivariant morphisms of log schemesX0 = Xn 99KX0 = X, where the underlying morphisms of schemes are normalized blow ups alongG-stable centers sitting overZ∪T, such that various properties of the log schemeXiwith the action ofGgradually improve to match all assertions of the Theorem. To simplify notation, we will, as a rule, replaceXwith the new log scheme after each step. The three conditions above will be achieved in three steps as follows.
4.1.1. Step 1. MakingX regular.— This is achieved by the saturated log blow up tower Ff(X, Z) :X0 99K X from Theorem3.4.9. In particular, the morphismX0 →Xis even log smooth. In the sequel, we assume that Xis regular, in particular, Z is a normal crossings divisor by exp.VI,1.7. We will callZ theboundary of X. In the sequel, these conditions will be preserved, so let us describe an appropriate restriction on further modifications.
4.1.2. Permissible blow ups. — After Step 1 any modification in the remaining tower will be of the form f: (X0, Z0)→(X, Z)whereX0 =BlV(X),Z0=f−1(Z∪V)andVhasnormal crossingswithZ, i.e. étale locally onXthere exist regular parameterst1, . . . , tdsuch thatZ=V(Ql
i=1ti)andV=V(ti1, . . . , tim). We call such modification of log schemespermissibleand use the convention that the centerVis part of the data. A blow up of schemesf: X0 → Xis calledpermissible(with respect to the boundaryZ) if it underlies a permissible modification. Since there is an obvious bijective correspondence between permissible modifications and blow ups we will freely pass from one to another. Note thatZ0 =fst(Z)∪E0, whereE0 =f−1(V)is the exceptional divisor.
4.1.3. Permissible towers. — Apermissible modification tower(Xd, Zd)99K(X0, Z0) = (X, Z)is defined in the obvious way and we say that a blow up towerXd99KXis permissible if it underlies such a modification tower.
Again, we will freely pass between permissible towers of these types. Note thatZi=Zsti ∪Ei, whereZsti is the strict transform ofZunderhi:Xi99KXandEiis the exceptional divisor ofhi(i.e. the union of the preimages of the centers ofhi).
Remark 4.1.4. — (i) Consider a permissible tower as above. It is well known that for anyione has thatXi
is regular,Zi is normal crossings andEiis even snc. For completeness, let us outline the proof. Both claims follow from the following: ifZis snc thenZiis snc. Indeed, the claim aboutEifollows by takingZ= ∅and the claim aboutZican be checked étale locally, so we can assume thatZis snc. Finally, ifZis snc thenZiis snc by Lemma4.2.9below.
(ii) Permissible towers are very common in embedded desingularization because they do not destroy reg-ularity of the ambient scheme and the normal crossings (or snc) property of the boundary (or accumulated exceptional divisor). Even when one starts with an empty boundary, a non-trivial boundary appears after the first step, and this restricts the choice of further centers. Actually, any known self-contained proof of embedded desingularization constructs a permissible resolution tower.
4.1.5. G-permissible towers. — In addition, we will only blow upG-equivariant centersV. So,f: X0=BlV(X)→ XisG-equivariant and the exceptional divisorE = f−1(V)is regular andG-equivariant and henceG-strict.
Such a blow up (or their tower) will be called G-permissible. It follows by induction that the exceptional divisor of such a tower isG-strict.
4.1.6. Step 2. MakingZsnc andG-strict. — Consider the stratification ofZby multiplicity: a pointz∈Zis in Zn if it has exactlynpreimages in the normalization ofZ. Note that{Zn}is precisely the log stratification as defined in3.3.1. Bydepthof the stratification we mean the maximaldsuch thatZd6= ∅. In particular,Zdis the only closed stratum. Step 2 proceeds as follows:Xi+1→Xiis the blow up along the closed stratum ofZsti. Remark 4.1.7. — What we use above is the standard algorithm that achieves the following two things: Z0is snc andZst=∅(see, for example, [de Jong, 1996, 7.2]). Even whenZis snc, the second property is often used in the embedded desingularization algorithms to get rid of the old components of the boundary.
4.1.8. Justification of Step 2. — Since the construction is well known, we just sketch the argument. First, observe thatZdhas normal crossings withZ, that is,X0 = BlZd(X)→Xis permissible. Thus,Z0 is normal crossings and henceZstis also normal crossings. A simple computation with blow up charts shows that the depth ofZst isd−1(for example, one can work étale locally, and then this follows from Lemma4.2.8below). It follows by induction that the tower produced by Step 2 is permissible, of lengthdand withZstd=∅. So,Zd= Edis snc by Remark4.1.4andG-strict by4.1.5.
4.1.9. Step 3. Making the inertia groups abelian and the action ofGonX\Zfree. — Recall (exp.VI,4.1) that the inertia strata are of the form XH = XH \S
H(H0XH0. Step 3 runs analogously to Step 2, but this time we will work with the inertia stratification ofXinstead of the log stratification, and will have to apply the same operation a few times. Let us first describe the blow up algorithm used in this step; its justification will be given in §4.2.
Letf{XH}: X0 99K Xdenote the following blow up tower. First we blow up the union of all closed strata XH. In other words,V0is the union of all non-empty minimalXH, i.e. non-emptyXHthat do not containXK with∅ ( XK ( XH. Next, we consider the family of all strict transforms ofXHand blow up the union of the non-empty minimal ones, etc. Obviously, the construction isG-equivariant. We will prove in Proposition 4.2.11thatf{XH} is permissible of length bounded by the length of the maximal chains of subgroups. Also, we will show in4.2.13thatf{XH} decreases all non-abelian inertia groups, so the algorithm of Step 3 goes as follows: until all inertia groups become abelian, applyf{XH}: X099KX(i.e. replace(X, Z)with(X0, Z0)).
4.2. Justification of Step 3. —
4.2.1. Weakly snc families. — Assume thatXis regular,Z,→Xis an snc divisor, and{Xi}i∈Iis a finite collection of closed subschemes ofX. For anyJ⊂Iwe denote byXJthe scheme-theoretic intersectionT
j∈JXj. The family {Xi}is calledweakly sncif eachXiis nowhere dense andXJis regular. The family{Xi}is calledweaklyZ-snc if it is weakly snc and eachXJ has normal crossings withZ. In particular,{Xi}i∈Iis weakly snc (resp. weakly Z-snc) if and only if the family{XJ}∅6=J⊂Iis weakly snc (resp. weaklyZ-snc).
Remark 4.2.2. — (i) Here is a standard criterion of being an snc divisor, which is often taken as a definition.
LetD,→Xbe a divisor with irreducible components{Di}i∈I. ThenDis snc if and only if the set of its irreducible components is weakly snc and each irreducible component ofDJis of codimension|J|inX.
(ii) The condition on the codimension is essential. For example,xy(x+y) =0defines a weakly snc but not snc family of irreducible divisors inA2k=Spec(k[x, y]).
(iii) The criterion from (i) implies that ifXis qe then the snc locus ofDis open—it is the complement of the union of singular loci ofDJ’s and the loci whereDJ is of codimension smaller than|J|. (Note that this makes sense for all points ofXbecauseDis snc at a pointx∈X\Dif and only ifX=D∅is regular atx.)
Lemma 4.2.3. — Let(X, Z)andGbe as achieved in Step 2. Then the family{XH}16=H⊂Gis weaklyZ-snc.
Proof. — Recall that for any subgroupH⊂Gthe fixed point subschemeXHis regular by Proposition exp.VI, 4.2, andXHis nowhere dense forH6=1by generic freeness of the action ofG. Since for any pair of subgroups K, H⊂Gwe have thatXH×XXK=XKH, the family is weakly snc. It remains to show thatY=XHhas normal crossings withZ. Note that it is enough to consider the case whenXis local with closed pointxandG =H.
The cotangent spaces atxwill be denotedT∗X = mX,x/m2X,x,T∗Y, etc. Their dual spaces will be called the tangent spaces, and denotedTX,TY, etc. Letφ∗:T∗XT∗Ydenote the natural map and letφ:TY,→TXdenote its dual. We will systematically use without mention that|G|is coprime to chark(x), in particular, the action of GonT∗Xis semi-simple.
The proof of exp. VI,4.2also shows that for any pointx ∈ XH, the tangent space Tx(XH)is isomorphic to(TxX)H. In particular,TY →∼ (TX)Gand henceφ∗maps(T∗X)G⊂T∗Xisomorphically ontoT∗Y. Therefore, U=Ker(φ∗)is theG-orthogonal complement to(T∗X)G, i.e. the onlyG-invariant subspace such that(T∗X)G⊕ U→∼ T∗X. LetZi =V(ti),1≤i≤nbe the components ofZand letdti ∈T∗Xdenote the image ofti. ByZ -strictness ofG, each lineLi=Span(dti)isG-invariant, soGacts ondtiby a characterχi. Without restriction of generality,χ1, . . . , χlfor some0≤l≤nare the only trivial characters. In particular,L=Span(dt1, . . . , dtn)is the direct sum ofLG=Span(dt1, . . . , dtl)and itsG-orthogonal complementL∩U, which (by uniqueness of the complement) coincides with Span(dtl+1, . . . , dtn). Complete the basis ofLto a basis{dt1, . . . , dtn, e1, . . . , em} ofT∗Xsuch that{dtl+1, . . . , dtn, e1, . . . , er}for somer≤mis a basis ofUand choose functionss1, . . . , smon Xso thatdsj=ejands1, . . . , srvanish onY. Clearly,t1, . . . , tn, s1, . . . , smis a regular family of parameters of OX,x, so the lemma would follow if we prove thatY=V(tl+1, . . . , tn, s1, . . . , sr).
SinceY is regular and Ker(φ∗)is spanned by the images oftl+1, . . . , tn, s1, . . . , sr, we should only check that these functions vanish onY. Thesj’s vanish onYby the construction, so we should check thattivanishes onYwheneverl < i≤n. Using the functorial definition from exp.VI,4.1of the subscheme of fixed points, we obtain thatZGi =Zi×XXG, henceZi×XYis regular by exp.VI,4.2. However,TYis contained inTZi, which is the vanishing space ofdti, hence we necessarily have thatY,→Zi.
4.2.4. Snc families. — A family of nowhere dense closed subschemes{Xi}i∈I is calledsnc (resp. Z-snc) at a pointxifXis regular atxand there exists a regular family of parameterstj∈OX,xsuch that in a neighborhood ofxeachXi(resp. and each irreducible componentZkofZ) passing throughxis given by the vanishing of a subfamilytj1, . . . , tjl. Note that the family{Xi}i∈IisZ-snc if and only if the union{Xi}∪{Zk}is snc. A family issncif it is so at any point ofX(in particular,Xis regular).
Remark 4.2.5. — (i) It is easy to see that the family{XH}H⊂Gis snc wheneverGis abelian. Indeed, it suffices to show that for any pointxthere exists a basis ofTxXsuch that eachTx(XH)is given by vanishing of some of the coordinates. But this is so because the action ofGonTxXis (geometrically) diagonalizable andTx(XH) = (TxX)H. In general, the family{XH}H⊂Gdoes not have to be snc, as the example of a dihedral groupDnwith n≥3acting on the plane shows.
(ii) IfZ,→Xis an snc divisor with componentsZiandV,→Xis a closed subscheme then the family{Zi, V}is snc if and only ifVhas normal crossings withZ.
The transversal case of the following lemma can be deduced from [ÉGAIV4§19.1], but we could not find the general case in the literature (although it seems very probable that it should have appeared somewhere).
Lemma 4.2.6. — Any weakly snc family with|I|=2is snc.
Proof. — We should prove that ifX=Spec(A)is a regular local scheme andY,Zare regular closed subschemes such thatT =Y×XZis regular then there exists a regular family of parameterst1, . . . , tn∈Asuch thatYand Zare given by vanishing of some set of these parameters. Letmbe the maximal ideal ofA, and letI,Jand K=I+Jbe the ideals definingY,ZandT, respectively. ByT∗X=m/m2,T∗Y, etc., we denote the cotangent
spaces at the closed point ofX. Note thatI/mI→∼ Ker(T∗X→T∗Y), and similar formulas hold forJ/mJand K/mK. Indeed, we can choose the parameters so thatY =V(t1, . . . , tl)and then the images oft1, . . . , tlform a basis both ofI/mIand Ker(T∗X→T∗Y).
Now, let us prove the lemma. Assume first thatY and Zare transversal, i.e. T∗X,→T∗Y ⊕T∗Z. Choose elementst1, . . . , tl+k such thatY = V(t1, . . . , tl), Z = V(tl+1, . . . , tl+k),l = codim(Y)andk = codim(Z). Then the imagesdti ∈ T∗X ofti are linearly independent because dt1, . . . , dtl span Ker(T∗X → T∗Y) and dtl+1, . . . , dtl+k span Ker(T∗X → T∗Z). Hence we can completeti’s to a regular family of parameters by choosingtl+k+1, . . . , tn such thatdt1, . . . , dtn is a basis ofT∗X. This proves the transversal case, and to es-tablish the general case it now suffices to show that ifYandZare not transversal then there exists an element t1∈m\m2which vanishes both onYandZ. (The we can replaceXwithX1=V(t1)and repeat this process untilYandZare transversal inXa =V(t1, . . . , ta).) Tensoring the exact sequence0→I∩J→I⊕J→K→0 withA/mwe obtain an exact sequence
(I∩J)/m(I∩J)→I/mI⊕J/mJ→φ K/mK→0
The failure of transversality is equivalent to non-injectivity ofφ, hence there exists an elementf∈I∩Jwith a non-zero image inI/mI⊕J/mJ. Thus,f∈m\m2and we are done.
4.2.7. Blowing up the minimal strata of a weakly snc family. — Given a weakly snc family{Xi}i∈I, we say that a schemeXJ withJ ⊂ Iisminimal if it is non-empty and anyXJ0 ( XJ is empty. Also, we will need the following notation: ifZ,→Xis a closed subscheme andD,→Xis a Cartier divisor with the corresponding ideals IZ,ID⊂OX, thenZ+Dis the closed subscheme defined by the idealIZID.
Proposition 4.2.8. — Assume thatXis regular,Z,→Xis an snc divisor with irreducible componentsZ1, . . . , Zl, and {Xi}i∈Iis aZ-snc (resp. weaklyZ-snc) family of subschemes. LetVbe the union of all non-empty minimal subschemes XJ,f:X0=BlV(X)→X,Xi0=fst(Xi)andZ0=f−1(Z∪V). Then
(i)X0is regular andZ0is snc.
(ii) The family{Xi0}i∈IisZ0-snc (resp. weaklyZ0-snc).
(iii) For anyJ⊂I, the scheme-theoretical intersectionXJ0=T
j∈JXj0coincides withfst(XJ).
(iv) For anyJ ⊂ I the total transformXJ ×XX0 is of the formXJ0 +DJ0 whereDJ0 is the divisor consisting of all connected components ofE0=f−1(V)contained inf−1(XJ).
Proof. — We start with the following lemma.
Lemma 4.2.9. — Assume thatXis regular,Zis an snc divisor andV,→Yare closed subschemes having normal crossings withZ. Letf:X0→Xbe the blow up alongV,Y0=fst(Y),Z0=f−1(Z∪V), andE0=f−1(V). ThenZ0is snc,Y0has normal crossings withZ0andY×XX0=Y0+E0.
Proof. — The proof is a usual local computation with charts. Take any pointu ∈ V and choose a regular family of parameterst1, . . . , tnatusuch thatY(resp.V, resp.Z) are given by the vanishing oft1, . . . , tm(resp.
t1, . . . , tl, resp.Q
i∈Iti), where0 ≤m≤l≤nandI⊂{1, . . . , n}. Locally overuthe blow up is covered byl charts, and the local coordinates on thei-th chart aretj0such thattj0 =tjforj > lorj=iandtj0 = ttj
iotherwise.
On this chart,Y×XX0(resp. Y0, resp. E0, resp. Z0) is given by the vanishing oft1, . . . , tm (resp. t10, . . . , tm0 , resp.ti0, resp.Q
j∈I∪{i}tj0), hence the lemma follows.
The lemma implies (i). In addition, it follows from the lemma thatfst(XJ)has normal crossings withZ0 andXJ×XX0 =fst(XJ) +DJ0. Thus, (iii) implies (iv), and (iii) implies (ii) in the case when the family{Xi}i∈I
is weaklyZ-snc. Note also that if this family isZ-snc then locally at any pointx∈ Xthere exists a family of regular parameterst={t1, . . . , tn}such that eachXJand each component ofZis given by the vanishing of a subfamily oftlocally in a neighborhood ofx. Then the same local computation as was used in the proof of Lemma4.2.9proves also claims (ii) and (iii) of the proposition. So, it remains to prove (iii) when the family is weakly snc. It suffices to prove that if (iii) holds forXJandXKthen it holds forXJ∪K. Moreover, (iii) does not involve the boundary so we can assume thatZ=∅. It remains to note that{XJ, XS}is an snc family by Lemma 4.2.6, hence our claim follows from the snc case.
4.2.10. Blow up tower of a weaklyZ-snc family. — LetXbe a regular scheme,Zbe an snc divisor and{Xi}i∈I
be a weaklyZ-snc family. By the blow up towerf{Xi}of{Xi}we mean the following tower: the first blow up h1 : X1 → X0 = Xis along the union of all non-empty minimal schemes of the formXJ for∅ 6= J ⊂I, the second blow up is along the union of all non-empty minimal schemes of the formhst1(XJ)for∅6=J⊂I, etc.
Proposition 4.2.11. — Keep the above notation. Then the towerf{Xi} is permissible with respect toZand its length equals to the maximal length of chains∅6= XJ1 ( · · · (XJd with∅6= Jd ( · · · (J1 ⊂I. Furthermore, the strict transform of any schemeXJis empty and the total transform ofXJis a Cartier divisor.
Proof. — The claim about the length is obvious. Letf{Xi} :Xd99KX0=Xand lethn :Xn 99KX0=Xbe itsn -th truncation. For eachi∈IsetXn,i=hstn(Xi)and for eachJ⊂IsetXn,J=T
i∈JXn,i. Using Proposition4.2.8 and straightforward induction on length, we obtain that the family{Xn,i}i∈Iis weaklyZn-snc,Xn,J=hstn(XJ), the blow upXn+1→Xnis along the union of non-empty minimalXn,J’s, andXJ×XXn =Xn,J+DJ,n, where DJ,nis a divisor. So, the tower is permissible, and sinceXd,J=∅we also have thatXJ×XXdis a divisor.
Remark 4.2.12. — We will not need this, but it is easy to deduce from the proposition that on the level of morphisms the modificationXd→Xis isomorphic to the blow up alongQ
∅6=J⊂IIXJ.
4.2.13. Justification of Step 3. — The blow up towerf{XH}: X099KXfrom Step 3 isG-equivariant in an obvious way, and it is permissible by Proposition4.2.11. In addition,Z0 = f−1(Z)∪f−1(S
16=H⊂GXH)and, sinceG acts freely onX0\f−1(S
16=H⊂GXH)→∼ X\S
16=H⊂GXH, it also acts freely onX0\Z0. It remains to show that applyingf{XH}we decrease all non-abelian inertia groups. Namely, for anyx0∈X0mapped tox∈Xwe want to show that eitherGxis abelian or the inclusionGx0(Gxis strict.
Let H ⊂ G be any non-abelian subgroup with commutatorK = [H, H]. Since XK×XX0 is a divisor by Proposition4.2.11, the universal property of blow ups implies thatX0 →Xfactors throughY = BlXK(X). On the other hand, it is proved in exp.VI,4.8thatY →Xis anH-equivariant blow up, and if a geometric point x → X with Gx = Hlifts to a geometric pointy → Y then Gy ( H. Therefore, the same is true for the G-equivariant modificationX0 →X, and we are done.