2000 Mathematics Subject Classification: 14B05, 32S25.The smooth equimultiple locus of embedded algebroid surfaces
appears naturally in many resolution processes, both classical and modern.
In this paper we explore how it changes by blowing–up.* Supported by FQM 304 and BFM 2000–1523.
** Supported by FQM 218 and BFM 2001–3207

Piedra-Sánchez, R.

Tornero, J.

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica Math. J. 30 (2004), 195–206EQUIMULTIPLE LOCUSOF EMBEDDED ALGEBROID SURFACESAND BLOWING–UP IN CHARACTERISTIC ZEROR. Piedra-Sa´nchez∗ and J. M. Tornero∗∗Communicated by V. Brˆınza˘nescuAbstract. The smooth equimultiple locus of embedded algebroid surfacesappears naturally in many resolution processes, both classical and modern.In this paper we explore how it changes by blowing–up.1. Introduction. During all this paper, we will consider K an al-gebraically closed field of characteristic 0 and S = Spec(K[[X,Y,Z]]/(F )) anembedded algebroid surface which, with no loss of generality, is considered to bedefined by a Weierstrass equationF (Z) = Zn +n−1∑k=0ak(X,Y )Zk,2000 Mathematics Subject Classification: 14B05, 32S25.Key words: Resolution of surface singularities, blowing–up, equimultiple locus.∗Supported by FQM 304 and BFM 2000–1523.∗∗Supported by FQM 218 and BFM 2001–3207.196 R. Piedra-Sa´nchez and J. M. Tornerowhere n is the multiplicity of S, that is ord(ak) ≥ n− k for all k = 0, . . . , n − 1.After the well–known Tchirnhausen transformation Z 7−→ Z − an−1/n we canget a Weierstrass equation of the formF (Z) = Zn +n−2∑k=0ak(X,Y )Zk.From now on, by a Weierstrass equation we will mean an equation likethis. Observe that, if we denote the initial form of F by F , the affine varietydefined by F (that is, the tangent cone of S) is a plane if and only if it is theplane Z = 0.In this situation the equimultiple locus of S isE(S) ={P ∈ Spec(K[[X,Y,Z]]) | F ∈ P (n)},which is never empty, as M = (X,Y,Z) always lies in E(S). Note that Z ∈ P forall P ∈ E(S).Geometrically speaking, the equimultiple locus represents points at wherethe multiplicity is the same as in the origin; hence they are the “closest” pointsto the origin in (coarse) terms of singularity complexity. We will denote by E0(S)the subset of smooth elements of E(S).Our aim is studying the set E(S) and, specifically, how its elements changeby blowing–up, in order to have a better understanding of the evolution of E0(S)through a resolution process. In this environment, our main result explains howcan we deduce “geometrically” E0(S(1))from E0(S), where S(1) is the result ofblowing–up S with center in an element of E0(S).The interest of this relies in the fact that the equimultiple locus con-tains important information for desingularization purposes. For instance, if Shas normal crossing singularities, blowing–up centers lying in E0(S) of maximaldimension resolves the singularity at the origin. This is the famous Levi–Zariskitheorem on the resolution of surface singularities, stated by Levi ([3]) and provedby Zariski ([11]).Concerning the extension of the results of this paper to the arbitrarycharacteristic case, we must say that, no matter which the characteristic is, theequimultiple locus has some very interesting properties, some of which we statebelow:(a) It is hyperplanar, that is, there exists a regular parameter which liesin every element of E(S). This was proved by Mulay ([4]) after some previouswork of Abhyankar ([2]) and Narasimhan ([5]).(b) As proved by Abhyankar ([1]), the Levi–Zariski theorem remains truein positive characteristic.Equimultiple locus of embedded algebroid surfaces. . . 197However, the techniques used cannot be applied to the general case, asit is not straightforward (actually, it is not known) that the same regular para-meter can define the tangent cone and a hyperplane containing the equimultiplelocus. In particular, Mulay’s proof of the hyperplanarity is not constructive, soa different approach may be needed to handle with characteristic p > 0 (see [7]for some instances).As for the extension of our result to higher dimensions is concerned, wecannot be very optimistic. First of all, there is the additional difficulty that,in positive characteristic, the equimultiple locus is not hyperplanar anymore,as shown by Narasimhan ([6]). Secondly, the Levi–Zariski resolution processas stated does not work, as proved by Spivakovski ([9]), which, in any case,makes results on this line less interesting. However, a less coarse version of theLevi–Zariski theorem for a more general type of varieties will surely mean agreat achievement and it will need results of this sort towards a more thoroughunderstanding of the evolution of the equimultiple locus by sucessive blowing–ups.2. Notation and technical results. For the sake of completeness,we recall here basic facts and technical results related to quadratic and monoidaltransformations that will be of some help in the sequel.For all what follows, let S be an embedded algebroid surface of multiplicityn,F = Zn +n−2∑k=0∑i,jaijkXiY jZk = Zn + n−2∑k=0ak(X,Y )Zka Weierstrass equation of S. We will denoteN(F ) ={(i, j, k) ∈ N3 | aijk 6= 0}.Definition. The elements of E(S) different from M will be called equi-multiple curves. The elements of E0(S) other than M will be called permittedcurves.Remark. The notion of permitted curves coincides with the one derivedfrom normal flatness in the work of Hironaka.Remark. In particular, note that we can assume P ∈ E0(S) to be, forinstance (Z,X), after a suitable change of variables in K[[X,Y ]]. Plainly, (Z,X)is permitted if and only if i+ k ≥ n for all (i, j, k) ∈ N(F ).Under these circumstances, the monoidal transform of S, centered in(X,Z), in the point corresponding to the direction (α : 0 : γ) (say α 6= 0) of198 R. Piedra-Sa´nchez and J. M. Tornerothe exceptional divisor is the surface S(1) defined by the equationF (1) =(Z1 +γα)n+∑(i,j,k)∈N(F )aijkXi+k−n1 Yj1(Z1 +γα)k.Observe that this only makes sense (that is, gives a non–unit) wheneverF (α, 0, γ) = 0. The homomorphismπP(α:0:γ) : K[[X,Y,Z]] −→ K[[X1, Y1, Z1]]X 7−→ X1Y 7−→ Y1Z 7−→ X1(Z1 +γα)will be called the homomorphism associated to the monoidal transformation in(α : 0 : γ) or, in short, the equations of the monoidal transformation. The overlineis because one must privilege a non-zero coordinate, but all the possibilities defineassociated equations.As for quadratic transforms (that is, blowing–ups with center M) is con-cerned: the quadratic transform of S in the point corresponding to the direction(α : β : γ) (say α 6= 0) of the exceptional divisor is the surface S(1) defined bythe equationF (1) =(Z1 +γα)n+∑(i,j,k)∈N(F )aijkXi+j+k−n1(Y1 +βα)j (Z1 +γα)k.Again this only makes sense whenever F (α, β, γ) = 0. Analogously, thehomomorphismπM(α:β:γ) : K[[X,Y,Z]] −→ K[[X1, Y1, Z1]]X 7−→ X1Y 7−→ X1(Y1 +βα)Z 7−→ X1(Z1 +γα)will be called the homomorphism associated to the quadratic transformation in(α : β : γ) or the equations of the quadratic transformation.Equimultiple locus of embedded algebroid surfaces. . . 199Remark. In the previous situation, consider a change of variables inK[[X,Y,Z]] given byϕ(X) = a1X′ + a2Y′ + a3Z′ + ϕ1(X′, Y ′, Z ′)ϕ(Y ) = b1X′ + b2Y′ + b3Z′ + ϕ2(X′, Y ′, Z ′)ϕ(Z) = c1X′ + c2Y′ + c3Z′ + ϕ3(X′, Y ′, Z ′),with ord (ϕi) ≥ 2.Assume also thatα = a1α′ + a2β′ + a3γ′β = b1α′ + b2β′ + b3γ′γ = c1α′ + c2β′ + c3γ′with (say) γ′ 6= 0. Then there is a unique change of variables ψ : K[[X1, Y1, Z1]] −→K[[X ′1, Y′1 , Z′1]] such thatψπM(α:β:γ) = πM(α′:β′:γ′)ϕ.Both this remark and its monoidal counterpart (which will not be used inthis paper) are easy, although rather long, so we skip the proofs. The interestedreader may consult [8] and [10] for the complete details.Definition. Let Q ∈ E(S), with Q = (Z,G(X,Y )). Then for u ∈P2(K), the ideal̟Mu (Q) =(Z1,πMu (G(X,Y ))Xord(G)1)is called the quadratic transform of Q in the point corresponding to u.Obviously, this definition makes sense only if the quadratic transform inthe direction u does. There is a natural version of monoidal transform of Q withcenter P , for all P ∈ E0(S).Notation. We will denote by ν the natural isomorphismν : K[[X,Y,Z]] −→ K [[X1, Y1, Z1]]sending X to X1, Y to Y1 and Z to Z1.3. The theorem. As our result is inspired by the resolution process, wewill restrict ourselves to the case which is interesting for desingularization issues:200 R. Piedra-Sa´nchez and J. M. Tornerothat where S and S(1) have the same multiplicity. In particular, this avoids somepossibilities.Lemma. If the tangent cone of S is not a plane, the multiplicity of anymonoidal transform of S is strictly less than n.P r o o f. If the tangent cone is not a plane, mind there is only one possibleelement in E0(S). After a suitable change of variables on K[[X,Y ]], let this curvebe (Z,X). Then F cannot depend on Y ,F = Zn +∑i+k=nai0kXiZk =n∏l=1(Z − αlX).That is, the directions of the exceptional divisor are (1 : 0 : αl) forl = 1, . . . , n; not all of them equal. Then, the equation for one of these monoidaltransforms isF (1) = (Z1 + αl0)n +∑aijkXi+k−n1 Yj1 (Z1 + αl0)kwhere we find the monomial ∏αl0 6=αl(αl0 − αl)Zm,with m = #{l | αl0 = αl}. This monomial cannot cancel in any case. So there isa monomial in F (1) of order strictly smaller than n and we are done. Theorem. Let S be an algebroid surface and S(1) a quadratic or monoidaltransform of S with the same multiplicity.(a) If S(1) is the monoidal transform of S with center P ∈ E0(S), then, eitherE0(S(1))= ν (E0(S)) or E0(S(1))= ν (E0(S) \ {P}).(b) Let S(1) be the quadratic transform of S in the point corresponding to u.Then:(b.1) If the tangent cone is not a plane then E0(S(1))= ν (E0(S)).(b.2) If the tangent cone is a plane, then in E0(S(1))we can find threetypes of curves:(i) The exceptional divisor of the transform.(ii) Primes ̟Mu (Q), with Q ∈ E(S) \ E0(S), which are tangent to theexceptional divisor.(iii) Primes ̟Mu (Q), with Q ∈ E0(S), where both ν(Q) and ̟Mu (Q) aretransversal to the exceptional divisor.Equimultiple locus of embedded algebroid surfaces. . . 201Moreover, if there is any prime of type (ii), it also appears the type (i)prime.P r o o f. We will do the proof case by case, although some arguments arecommon to various instances. In what follows let F be, as usual, a Weierstrassequation of S.Case (a)From the previous lemma, we can assume that the tangent cone is theplane Z = 0. The basic tool for this situation is the following:Remark. A monoidal transformation which does not imply a descentof the multiplicity cannot create new permitted curves.This can be proved easily as follows: under the hypothesis of case (a), let(Z,G) be a permitted curve. Then S(1), the monoidal transform of S with centerat (Z,G), is given byF (1) = Zn1 +n−2∑k=0ak(X1, Y1)G(X1, Y1)n−kZk1 .This consists simply on taking (Z,G) to (Z,X) by a change of variables(say ϕ) on K[[X,Y ]], applying the transform (the only point in the exceptionaldivisor in this case is the point corresponding to the direction (1 : 0 : 0)) andtaking ϕ−1 on K[[X1, Y1]]. The result follows directly.This remark clearly implies case (a) of the theorem.Case (b.2)Some arguments in this case will be used for the other, so we will beginfor it. Let us start for the direction (1 : 0 : 0) (the direction (0 : 1 : 0) is obviouslysymmetric). IfF = Zn +∑(i,j,k)∈N(F )aijkXiY jZk,thenF (1) = Zn1 +∑(i,j,k)∈N(F )aijkXi+j+k−n1 Yj1 Zk1 .Note that, as F is a Weierstrass equation and the multiplicity does notchange, F (1) is a Weierstrass equation for S(1), hence all the elements of E(S(1))are contained in Z1 = 0, therefore all permitted curves in E(S(1))can be assumedto be of the form P = (Z1, γX1 + δY1 +G(X1, Y1)), with ord(G) ≥ 2.Let us prove now that, if a permitted curve transversal to the exceptionaldivisor appears in E0(S(1)), it comes from a permitted curve in E0(S) which wasalso transversal to the exceptional divisor (up to the action of ν).202 R. Piedra-Sa´nchez and J. M. TorneroSuppose we have such a curve (that is, a prime as above with δ 6=0). Then, applying the Weierstrass Preparation Theorem, we may write P as(Z1, Y1 +H(X1)). We have the diagramK[[X,Y,Z]] K[[X ′, Y ′, Z ′]]K[[X1, Y1, Z1]] K[[X′1, Y′1 , Z′1]]--? ?ϕψpiM(1:0:0)piM(1:0:0)with changes of variablesϕ(X) = X ′ϕ(Y ) = Y ′ −X ′H(X ′)ϕ(Z) = Z ′,ψ(X1) = X′1ψ(Y1) = Y′1 −H(X′1)ψ(Z1) = Z′1So, looking at the right vertical arrow, we have found a quadratic trans-form on the direction (1 : 0 : 0) which gives rise to the permitted curve (Z ′1, Y′1).This clearly implies that (Z ′, Y ′) was permitted in S. This proves the assertion.Another way of seeing this is saying that, if there were no permitted curveswhich were transversal to the exceptional divisor, all permitted curves after theblowing–up must be tangent to it.Let us prove now that, if (Z1,X1+Ys1 v(Y1)) with s > 1, appears, so does(Z1,X1). WriteF (1) = Zn1 +n−2∑k=0a(1)k (X1, Y1)Zk1 ,where it must hold a(1)k = (X1 + Ys1 v(Y1))n−kb(1)k (X1, Y1).Fix then k0 ∈ {0, . . . , n − 2} and choose from all monomials in b(1)k0theminimal one for the lexicographic ordering, say Xi01 Yj01 . Then all monomialsappearing in a(1)k0have exponent in X1 greater or equal than i0 and, besides, themonomial Xi01 Yj0+s(n−k0)1 actually appears in a(1)k0, as it cannot be cancelled.Now it is plain that (i, j, k) ∈ N(F (1))if and only if (i− j− k+n, j, k) ∈N(F ), soi0 ≥ s (n− k0) + k0 − n ≥ n− k0.Hence (Z1,X1) ∈ E(S(1)).Equimultiple locus of embedded algebroid surfaces. . . 203Let us prove then the existence, in this case, of the curve Q on E(S)announced in the theorem. As previously, we will consider α = 0. Note G(Y1) =Y 21 v (Y1). We will prove that there exists a power series H(X,Y ) verifying:(1) ord(H) = ord(G) = λ > 1.(2) H is regular on Y of order λ.(3) There is a unit u(X1, Y1) such that1Xλ1H(X1,X1Y1) = u(X1, Y1)(X1 +G(Y1)).This implies (quite straightforwardly) that F ∈ Q(n) and (Z1,X1 +G (Y1))= πM(1:0:0)(Q), with Q = (Z,H(X,Y )). The second part is trivial and, for the firstpart it is enough proving(X1 +G(Y1))n−k |a(1)k (X1, Y1) =⇒ H(X,Y )n−k|ak(X,Y ),for k = 0, . . . , n − 2. Assume it is not so; then by the Weierstrass PreparationTheorem and being H regular with respect to Y , we can writeak(X,Y ) = q(X,Y )H(X,Y )n−k +(n−k)λ−1∑j=0σj(X)Yj .Now we apply πM(1:0:0) and we obtainXa1a(1)k (X1, Y1)n−k == Xb1q′ (X1, Y1) (X1 +G(Y1))n−k +(n−k)λ−1∑j=0(σj(X1)Xj1)Y j1 .Now, as (X1 +G(Y1))n−k divides a(1)k (X1, Y1) it also must divideXa1a(1)k (X1, Y1), hence the uniqueness of quotient and remainder in the Weier-strass Preparation Theorem implyσj (X1)Xj1 = 0, for all j = 0, . . . , (n − k)λ− 1,and subsequently H(X,Y )n−k|ak(X,Y ).So let us prove the existence of H and u. Write up X1 +G(Y1) asX1 +G(Y1) = X1 +∑i≥λαiYi1 ,204 R. Piedra-Sa´nchez and J. M. Torneroand the power series we look for asH(X1, Y1) =∑i+j=k≥λβijXi1Yj1 , u(X1, Y1) =∑i+j=k≥0γijXi1Yj1 .It must hold∑i+j=k≥λβijXk−λ1 Yj1 = ∑i+j=kγijXi1Yj1X1 +∑i≥λαiYi1 ,which, for order 0, amounts toβλ,0 = γ0,00 = 0.On the other hand, for order 1 we haveβλ+1,0X1 + βλ−1,1Y1 = γ0,0X1;that is, βλ−1,1 = 0 and βλ+1,0 = γ0,0, whose value can be taken to be 1.As for order 2,βλ+2,0X21 + βλ,1X1Y1 + βλ−2,2Y21 = γ0,0α2Y21 + γ1,0X21 + γ0,1X1Y1,which forces βλ−2,2 = α2 and let us freedom for fixing βλ+2,0 = γ1,0, and βλ,1 =γ0,1.Observe then the following facts:• Each βij appears only for order i+ 2j − λ.• In order k, all γij with i + j < k appear, but they never have relationsamong them, only those of the typeβab =∑γcdαe.Therefore it is clear that we can choose arbitrarily the γij , and this choicedetermines the βij . Therefore both H and u exist.It only remains proving that H can be chosen such that H(0, Y ) has orderλ. But this is direct from the formula for order λ;β0,λYλ1 = γ0,0αλYλ1 6= 0,so β0,λ 6= 0.Equimultiple locus of embedded algebroid surfaces. . . 205For the results at points (1 : α : 0) it suffices considering the (commuta-tive) diagramK[[X,Y,Z]] K[[X ′, Y ′, Z ′]]K[[X1, Y1, Z1]] K[[X′1, Y′1 , Z′1]]--? ?ϕψpiM(1:α:0)piM(1:0:0)with ϕ given by ϕ(X) = X ′ϕ(Y ) = Y ′ − αX ′ϕ(Z) = Z ′Remark. This case (b.2), in geometrical terms, may be expressed asfollows:• Permitted curves transversal to the exceptional divisor cannot be creatednor erased.• Permitted curves tangent to the exceptional divisor are erased.• Permitted curves tangent to the exceptional divisor can be created fromdesingularization of equimultiple (singular) curves. In this case, one ofthem must be the exceptional divisor itself.Case (b.1)Remark. In the conditions of (b.1), let P = (α : β : γ) a pointin the tangent cone with multiplicity r. Then, the quadratic transform of Son (α : β : γ) has, at most, multiplicity r. This is straightforward, using, forinstance, the Taylor expansion of F .So we only need to be concerned about points of multiplicity n on thetangent cone. Changing the variables if needed we may consider that the pointis (0 : 1 : 0) and, subsequently, F does not depend on Y .We will first prove that the quadratic transform cannot have permittedcurves. Note that, in (b.2), we have showed that, if a new permitted curveappears, so does the exceptional divisor (and we did not use the fact that F = Znfor proving this). But (Z, Y ) cannot be a permitted curve, F (1) having monomialsin K[X,Z] other than Zn.206 R. Piedra-Sa´nchez and J. M. TorneroNow we explain why the quadratic transform does not erase permittedcurves either. In fact if there is a permitted curve (only one is possible), we maytake it to be (Z,X), after the customary change of variables. Then it is plainthat, after a quadratic transform on the direction (0 : 1 : 0), (Z1,X1) remainspermitted, simply looking at the characterization given in the previous section.This finishes the proof of the theorem.REFERENCES[1] S. S. Abhyankar. Resolution of Singularities of Embedded Algebraic Sur-faces. Academic Press, New York, 1966.[2] S. S. Abhyankar. Good points of a hypersurface. Adv. Math. 68 (1988),87–256.[3] B. Levi. Risoluzione delle singolarita` puntualli delle superficie algebriche.Atti Accad. Sci. Torino 33 (1897), 66–86.[4] S. B. Mulay. Equimultiplicity and hyperplanarity. Proc. Amer. Math. Soc.89 (1983), 407–413.[5] R. Narasimhan. Hyperplanarity of the equimultiple locus. Proc. Amer.Math. Soc. 87 (1983), 403–408.[6] R. Narasimhan. Monomial equimultiple curves in positive characteristic.Proc. Amer. Math. Soc. 89 (1983), 402–406.[7] R. Piedra. Estudio local de singularidades de superficies sobre cuerpos decaracter´ıstica arbitraria. Ph. D. Thesis, Univ. de Sevilla, 1978.[8] T. Sa´nchez Giralda. Teor´ıa de singularidades de superficies algebroidessumergidas. Ph. D. Thesis, Univ. Complutense, 1976.[9] M. Spivakovski. A counterexample to the theorem of Beppo Levi in threedimensions. Invent. Math. 96 (1989), 181–183.[10] J. M. Tornero. Aspectos locales de singularidades de superficies: Superfi-cies de Puiseux. Ph. D. Thesis, Univ. de Sevilla, 2001.[11] O. Zariski. Reduction of singularities of algebraic three dimensional vari-eties. Ann. of Math. 45 (1944), 472–542.Departamento de A´lgebraUniversidad de Sevilla.Apdo 1160, 41080 Sevilla, Spaine-mail: piedra@algebra.us.ese-mail: tornero@algebra.us.es Received March 22, 2004

Equimultiple Locus of Embedded Algebroid Surfaces and Blowing–up in Characteristic Zero

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Equimultiple Locus of Embedded Algebroid Surfaces and Blowing–up in Characteristic Zero

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