On ν-quasiordinary surface singularities and their resolution

Mediterr. J. Math. (2024) 21:168 https://doi.org/10.1007/s00009-024-02709-x 1660-5446/24/050001-17 published online August 5, 2024 c The Author(s) 2024 On ν-Quasiordinary Surface Singularities and Their Resolution Fuensanta Aroca and Jos´eM.Tornero Abstract. Quasiordinary power series were introduced by Jung at the beginning of the 20th century, and were not paid much attention until the work of Lipman and, later on, Gao. They have been thoroughly studied since, as they form a very interesting family of singular varieties, whose properties (or at least many of them) can be encoded in a discrete set of integers, much as what happens with curves. Hironaka proposed a generalization of this concept, the so-called ν-quasiordinary power series, which has not been examined in the literature in such detailed way. This paper explores the behavior of these series under the resolution process in the surface case. Mathematics Subject Classification. 14H20, 32S25. Keywords. Resolution of surface singularities, ν-quasiordinary singularities, characteristic exponents. 1. Introduction: Definitions and Tools Let kbe an algebraically closed field of characteristic zero, m>1 an integer. Let us write R=kX,Y ,S=kX1/m,Y1/m. We will call Sa ring of Puiseux power series and we can consider m fixed, so we will not be writing (as maybe we should for full precision) Sm. Let K=Q(R)andL=Q(S) be the respective quotient fields. We will write ord(·) for the usual order of a power series (that is, the smallest degree of non-zero monomials of the series). We will recall the concept of ν-quasiordinary power series (in two variables) which was introduced by Hironaka. Definition 1. Let ζX1/m,Y1/m= (i,j)∈Z2 ≥0 zijXi/mYj/m ∈S. 168 Page 2 of 17 F. Aroca and J. M. Tornero MJOM The series ζis said to be ν-quasiordinary (ν-qo in the sequel) if it can be written as ζ=Xa/mYb/muX1/m,Y1/m,with u(0,0) =0. In this situation Xa/mYb/m will be called the monomial factor of ζand (a/m, b/m) will be called the monomial exponent of ζ. An algebroid surface Hthat can be defined by a Weierstrass polynomial F(Z)=Zn+ n−1  i=0 ai(X,Y )Zi,with ai(X,Y )∈R, such that all of its roots in Lare ν-qo series is called a ν-quasiordinary surface. Analogously, F(Z) will be then called a ν-quasiordinary Weierstrass polynomial. Remark 1.The name ν-qo is inherited from a well-known family of singularities, the so-called quasiordinary power series (qo in what follows), whose origin goes back to the work of Jung [5]. They were studied essentially after Lipman devoted his Ph.D. Thesis to their properties [7]. A qo power series η∈Ris the root of a Weierstrass polynomial F(Z) whose discriminant has the form Δ(F)=XaYbu(X,Y ),with u(0,0) =0. The Jung–Abhyankar Theorem [5] implies, among other things, that η is a ν-qo series. But qo series have some additional properties which might not be shared by general ν-qo series. For example, ζ=X2/3Y1/31+X1/3+Y1/3 is a ν-qo power series whose minimal polynomial F(Z) verifies Δ(F)=X48Y24(X+Y)6u(X, Y ), and therefore is not qo. The number of references, starting from the seminal works of Lipman and Gau [3,7] devoted to qo series and their resolution is too big to fit this article (even using the margins). For a given ζ∈S, the set Γ(ζ)=(i, j)∈Z2 ≥0|zij =0  will be called the cloud of exponents of ζ(mind they are not the actual exponents). Equivalently, ζis ν-qo if and only if min <X Γ(ζ)min <Y Γ(ζ)=∅, where <Xis the partial ordering that only takes into account the Xcoordinate (and likewise for <Y). Also, speaking in combinatorial terms, as ζis ν-qo if and only if there exists (a, b)∈Γ(ζ) (corresponding to the monomial factor) such that Γ(ζ)⊂(a, b)+Z2 ≥0. MJOM On ν-Quasiordinary Surface Singularities Page 3 of 17 168 If F(Z) is irreducible, being ν-qo is equivalent to being the minimal polynomial of a ν-qo series. If the conjugates of ζ(including itself) are then ζ1,...,ζ n, the Newton diagram of F(noted N(F)) can be seen as the Minkowski sum of the Newton diagrams of the polynomials Z−ζi∈R[Z]fori=1,...,n (essentially all the possible sums of npoints, taking one from each N(Z−ζi)[6]). Either using this or by simply writing down the monomials in F(in terms of the ζi’s) it is direct to notice that N(F) only has two proper vertices: (0,0,n), corresponding to the leading monomial, and (an/m, bn/m, 0) (the product of the monomial factors of ζ1,...,ζ n). Definition 2. Let ζ∈S. We will call {(i1,j 1),...,(is,j s)}⊂Γ(ζ) a set of distinguished exponents of ζif K(ζ)=KXi1/mYj1/m,...,Xis/mYjs/m. Remark 2.The definition of distinguished exponents can of course be made for any number of variables, and a finite set of distinguished exponents always exists [10] (for any number of variables). The Puiseux pairs of an algebroid curve and the characteristic pairs of a qo surface are examples of distinguished exponents. In the above situation, [K(ζ):K] is precisely m2/g, where gis the gcd of the 2 ×2-minors of the matrix [10] M=m0i1... i s 0mj 1... j s For the sequel, we will write this as g= (2) gcd(M). If ζis ν-qo we can assume (i1,j 1) is the pair corresponding to the monomial factor of ζ. For this work, we will restrict ourselves now and then to irreducible ν-qo polynomials (or surfaces), as a reducible surface is the union of irreducible ones, and such union will be ν-qo if and only if all of its irreducible components are. Remark 3.When working with a given ζ∈Sand its minimal polynomial F(Z), we can always consider that Γ(ζ)∩(Zm)2=∅, that is, there are no monomials in ζwhich lie also in R. This can be achieved by performing the Tschirnhausen transformation on F(Z), that is, if F(Z)=Zn+an−1(X,Y )Zn−1+···+a0(X, Y ), then the change of variables Z−→ Z−1 nan−1(X,Y ) will take F(Z) to the minimal polynomial of the power series resulting from erasing all the monomials in ζwith exponents in (Zm)2. This clearly preserves the ν-qo character. 168 Page 4 of 17 F. Aroca and J. M. Tornero MJOM So, from now on, we will assume that all the power series we are working with have no exponents in (Zm)2or, equivalently, an−1(X, Y )=0. For the sequel we will need to distinguish between two situations: the transversal and the non-transversal one, which will need different strategies to be dealt with. Definition 3. A power series ζis called transversal if ord(ζ)≥1. If this is not the case, ζis called non-transversal. Its minimal polynomials are named accordingly. Equivalently, with the previous notations, ζis transversal if and only if ord(F) = degZ(F)=n. The goal of the next sections, following [7], is to show that the property of being a ν-qo power series is stable by blowing-up. As we are interested mainly in the resolution process, we will first take into account those blowingups whose centers are permitted, in the sense of Hironaka [4]. That is, the center can be: •The origin. •A regular equimultiple curve (if there is any). Following classical nomenclature, blowing-up the origin will be called a quadratic transformation, while blowing-up a permitted curve will be called a monoidal transformation. These transformations are best studied separately. 2. Monoidal Transformations Monoidal transformations, as noted above, are blowing-ups centered in a permitted curve. And the existence of permitted curves implies the following result, which allows us to simplify the casuistry. A qo version is available in [7] and a version in the context of surfaces defined by Puiseux power series can be found in [9]. We include part of the latter proof as it is necessary to establish the ν-qo character. Proposition 1. Let Hbe a ν-qo irreducible surface defined by ζ=Xa/mYb/muX1/m,Y1/m∈S, and F(Z)the minimal polynomial of ζ.Then: (a) If ord(ζ)=(a+b)/m ≥1, a prime ideal (Z,c(X,Y )) lies in the equimultiple locus if and only if c(X, Y )|ζ. (b) If ord(ζ)=(a+b)/m < 1either there are no curves in the equimultiple locus or there exists a change of variables which takes the surface into aν-qo transversal equation. Proof. The first part is proved verbatim in [9] (for all Puiseux power series, not necessarily ν-qo). The second part consists actually of the mentioned proof in [9], together with a straightforward remark, but some details must be given to establish the result properly. Write then F(Z), the minimal polynomial of ζ∈Sas F(Z)=Zn+an−2(X,Y )Zn−2+···+a0(X, Y ), MJOM On ν-Quasiordinary Surface Singularities Page 5 of 17 168 with ord(F)<n. We look up first for equimultiple curves plying in Z=0,whichmust then be of the form (Z, c(X,Y )). It is fairly easy then to realize that equimulitplicity implies c(X,Y )|ζ(details can be seen in [9]), which is impossible since ord(ζ)<1andc(X,Y )∈R. Let us then look for curves pnot lying in Z= 0. If the initial form of ζ contains both X1/m and Y1/m it comes out that both Xand Yappear in F, the initial form of F. Then, as D(F)∈p, for any derivation Dof order l−1, it is plain that p=(X,Y ), which is impossible since Fis a monic polynomial in Z. Hence ζcanbeassumedtobe ζ=Xa/m +ζ0,with a<m, ord(ζ0)>ord(ζ). But, if ζ0contained a monomial of the form Xb/mYc/m with b<a,we can take one of these monomials with minimal b, and it will follow (for the same reason as above) that p=(X,Y ). So, if there are equimultiple curves not lying in Z= 0, the series must have the form ζ=Xμ/mu(X1/m,Y1/m), and, therefore, (X,Z) is an equimultiple curve as well. Let us prove then that in this case there is a ν-qo transversal power series η∈Sdefining H. We know that Fis regular with respect to Zand X. So, let us call  Fthe Weierstrass polynomial associated to Fwith respect to X. Let now Tbe a new variable. As the units of Rhave all of their μ-th roots inside R, the power series Xμu(X,Y )−Tμ∈kX,Y, T has a factor Xu(X,Y )−T, where uis an μ-th root of u. By the Weierstrass Preparation Theorem, there exists a unit u(X,Y, T) such that u(X,Y, T)·(Xu(X,Y )−T)=X−R(Y,T) Doing X= 0 we can easily find R(Y,T)=Tu(0,Y,T). Define then G(Y,T)=u(0,Y,T), which is a unit in kY,T. In this situation, in [9] it is proved that η=Zm/μGmY1/m,Z1/μis a transversal Puiseux power series which is a root of  F. And it is easy to see it is ν-qo, as G(Y,T) is a unit.  Now, if we can blow-up a (permitted) curve we might as well assume: •We are in the transversal case •Our curve is given by p=(Z,X)orbyq=(Z,Y ). We can also assume that, in fact, F=Zn, as otherwise Fwill be reducible. This is a classical result but we have been unable to find an easily accessible reference, so we have included one in the “Appendix”. 168 Page 6 of 17 F. Aroca and J. M. Tornero MJOM Theorem 1. Let Hbe a ν-qo irreducible surface defined by ζ∈Sand F(Z) the minimal polynomial of ζ, assuming F=Zn.Letalsop=(Z, X)be a permitted curve. Then, the monoidal transform of Fcentered in pis the ν-qo equation defined by ζ/X. the analogous result holds for q=(Z,Y ). Proof. Everything is almost done, as in this case the monoidal transform has a unique equation given by F(1)(Z)=Zn+ n−2  i=0 ai(X,Y ) Xn−kZi. The result is then clear, as X|ζand hence, as c(X,Y )∈R, it also divides all of its conjugates. F(1)(Z) is then the minimal polynomial of ζ/X and hence irreducible.  3. Quadratic Transformations (I): The Transversal Case Moving on to the blowing-ups which are centered in the origin, we will consider different cases, depending on the form of tangent cone, as the strategies needed are quite different. 3.1. The Case F(Z)=Zn As above, assume we have F(Z), the minimal polynomial of a ν-qo power series ζX1/m,Y1/m∈S, written as F(Z)=Zn+an−1(X,Y )Zn−1+···+a0(X, Y ) = n  i=1 Z−ζiX1/m,Y1/m, where {ζ1,...,ζ n}is the set of conjugates of ζ(including itself). When the tangent cone is the plane Z= 0, the quadratic transformations have infinitely many local equations, each one corresponding to a direction of the exceptional divisor. In this case, these directions are (1 : α: 0), for α∈kand (0 : 1 : 0). Consider first the case of direction (0:1:0).Letλ=ord(ζ)>1. Then the equation of the quadratic transform is F(1) =1 YnF(XY,Y,YZ)= 1 Yn n  i=1 YZ−ζiX1/mY1/m,Y1/m, and therefore F(1) = n  i=1 Z−1 YζiX1/mY1/m,Y1/m, where it should be noted that ζiX1/mY1/m,Y1/m/Y ∈S, because we are in the transversal case and it is, in fact, ν-qo. MJOM On ν-Quasiordinary Surface Singularities Page 7 of 17 168 Let us see what can be said concerning the irreducibility of F(1)(Z). If we write ζ= (i,j)∈Z2 ≥0 zijXi/mYj/m, then the corresponding root of F(1)(Z) is, precisely, ζ(1) = (i,j)∈Z2 ≥0 zijX(i+j−m)/mYj/m. If we prove that ζ(1) has, at least, nconjugates, then F(1) will be proved to be irreducible (and, in fact, to be the minimal polynomial of ζ(1)). Let then be {(i1,j 1),...,(is,j s)}⊂Γ(ζ) a set of distinguished exponents of ζ. Therefore any (i, j)∈Γ(ζ) verifies a relationship (i, j)=λ1(i1,j 1)+···+λs(is,j s)mod(Zm)2, for some λ1,...,λ s∈Z. Now all exponents of ζ(1) are precisely those of the form (i+j−m, j) for some (i, j)∈Γ(ζ), so such an exponent must verify (i+j−m, j)=λ1(i1+j1−m, j1)+···+λs(is+js−m, js)mod(Zm)2, which proves that {(i1+j1−m, j1),...,(is+js−m, js)}⊂Γζ(1) is a set of distinguished exponents of ζ(1). Now, given such a set of distinguished exponents, we know that the number of conjugates of ζ(1) is m2/g, with g= (2) gcd M(1), where M(1) =m0i1+j1−m ... i s+js−m 0mj 1... j s. As this matrix can be transformed into M=m0i1... i s 0mj 1... j s by means of elementary transformations (adding rows/columns), said (2) gcd is the same for Mand M(1),soζ(1) has as many conjugates as ζ,thatis, exactly nof them. Therefore F(1) is the minimal polynomial of ζ(1). So, a quadratic transformation in the direction (0 : 1 : 0) preserve both the ν-qo condition and the irreducibility. Clearly the same goes for the direction (1:0:0)andthereforeitonlyreststostudywhathappenswith quadratic transformations when we choose a direction (1 : α: 0), for α∈k∗. First, remember that an equation of the transform F(1) in this case is F(1) =1 XnF(X,X(Y+α),XZ). 168 Page 8 of 17 F. Aroca and J. M. Tornero MJOM As all units in Rhave n-th roots, if we write Y+α=ξ(Y)n,wecan parallel the above argument which shows that the roots of F(1) are then ζ(1) 1=1 Xζ1X1/m,X1/mξ(Y),...,ζ(1) n=1 XζnX1/m,X1/mξ(Y), which are ν-qo series (or units). To see this, mind that, if ζ=Xa/mYb/m ⎛ ⎝γ+ i+j>0 aijXi/mYj/m⎞ ⎠, for some γ= 0, then ζ(1) =X(a+b−m)/mξ(Y)b⎛ ⎝γ+ i+j>0 aijX(i+j)/mξ(Y)j⎞ ⎠, and the second and third factors are in fact units, while the first is not, as λ=ord(ζ)>1. However, irreducibility poses a more difficult aspect here. Example 1.Let us consider F(Z)=Z2−X3Y, which is clearly ν-qo (and qo in fact), and let us choose the direction (1 : 1 : 0). Then the resulting equation is F(1)(Z)=Z2−X2(Y+1), which is the product of two linear factors. Let us see when and why we lose the irreducibility of the equation. Let us write, as previously, ζX1/m,Y1/m= (i,j)∈Z2 ≥0 zijXi/mYj/m,Y+a=ξ(Y)n. Then we get ζ(1) =1 XζX1/m,X1/mξ(Y)= (i,j)∈Z2 ≥0 zijX(i+j−m)/mξ(Y)j, that is, Γζ(1)=(i+j−m, km)|(i, j)∈Γ(ζ),k∈Z≥0. In particular, if {(i1,j 1),...,(is,j s)}⊂Γ(ζ) was a set of distinguished exponents, then so it is {(i1+j1,0) ,...,(is+js,0)}⊂Γζ(1), and then the number of conjugates of ζ(1) is m2/g, where gis the (2) gcd of M(1) =m0i1+j1... i s+js 0m0... 0 MJOM On ν-Quasiordinary Surface Singularities Page 9 of 17 168 This shows that whether the irreducibility is preserved or not (that is, whether the number of conjugates remain the same after the quadratic transformation) only depends on ζor, if we want to restrict ourselves to a finite set of data, on a set of distinguished exponents. In particular, it does not depend on α. 3.2. The Case F(Z)=Zn So we move now to the case where the tangent cone is no longer a plane. We preserve the notations from the previous case. Now, in particular, as we are in the transversal case, we must have that ord(ζ)=1and F= n  i=1 Z−ζi, which implies that, in the initial form of Fthere must be some monomials from a0(X,Y ). In particular, (0 : 0 : 1) is not an allowed direction and we have to look at local equations in, say, (1 : α:β). Analogously one may deal with directions (γ:1:δ) if necessary. Write as before Y+α=ξ(Y)n. An equation of F(1) in this direction is then F(1) =1 XnF(X,X(Y+α),X(Z+β)) = n  i=1 Z+β−1 XζiX1/m,X1/mξY1/m Mind that some of the series ζ(1) i=−β+1 XζiX1/m,X1/mξY1/m might be units. In fact, there are exactly as many non-units as indicated by the multiplicity of (1 : α:β)inF[2]. These series are clearly ν-qo. The arguments on irreducibility are exactly parallel to those of the case F=Zn. The results in this section can be summarized as follows: Theorem 2. Let Hbe a ν-qo irreducible surface defined by a transversal ν-qo ζ∈Sand F(Z)the minimal polynomial of ζ.Then: •If we perform a quadratic transformation in the directions (1:0:0)or (0:1:0)we obtain a ν-qo irreducible surface. •If we perform a quadratic transformation in a direction other than (1 : 0:0)or (0:1:0)we obtain a ν-qo surface which might be reducible. Whether the resulting surface is reducible or not does not depend on the direction chosen for the transformation. More specifically, if depends only on ζ(or on any set of distinguished exponents of ζ). 168 Page 16 of 17 F. Aroca and J. M. Tornero MJOM Author contributions Both authors have worked in the manuscript. Funding Funding for open access publishing: Universidad de Sevilla/CBUA. Data availability No datasets were generated or analyzed during the current study. Declarations Conflict of interest The authors declare no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References [1] Aroca, F., Cano, J.: Formal solutions of linear PDEs and convex polyhedra. J. Symb. Comput. 32, 717–737 (2001) [2] Cobo, H., Soto, M.J., Tornero, J.M.: Combinatorics and their evolution in resolution of embedded algebroid surfaces. Asian J. Math. 24, 725–758 (2020) [3] Gau, Y.-N.: Embedded topological classification of quasi-ordinary singularities. Topology 25, 495–519 (1986) [4] Hironaka, H.: Desingularization of excellent surfaces. In: Cossart, V., Giraud, J., Orbanz, U. (eds.) Resolution of Surface Singularities. Lecture Notes in Mathematics, vol. 1101, pp. 99–132 (1984) [5] Jung, H.W.E.: Darstellung der Funktionen eines algebraischen K¨orpers zweier unabh¨angigen Ver¨anderlichen x,yin der Umgebung x=a,y=b.J.Reine Angew. Math. 133, 289–314 (1908) [6] Khovanskii, A.: Three lectures on Newton polyhedra. S¯urikaisekikenky¯usho K¯oky¯uroku 1233, 1–17 (2001) [7] Lipman, J.: Quasi-ordinary singularities of embedded surfaces. Ph.D. Thesis, Harvard Univ. (1965) [8] S´anchez-Giralda, T.: Teor´ıa de singularidades de superficies algebroides sumergidas. Ph.D. Thesis, Univ. Complutense (1974) [9] Tornero, J.M.: Some geometric aspects of Puiseux surfaces. Rev. Mat. Iberoamericana 19, 731–744 (2003) MJOM On ν-Quasiordinary Surface Singularities Page 17 of 17 168 [10] Tornero, J.M.: On Kummer extensions of the power series field. Math. Nach. 281, 1511–1519 (2008) Fuensanta Aroca Instituto de Matem´aticas, Unidad Oaxaca Universidad Nacional Aut´onoma de M´exico (UNAM) Oaxaca Mexico e-mail: [email protected] Jos´eM.Tornero Departamento de ´ Algebra and IMUS Universidad de Sevilla Seville Spain e-mail: [email protected] Received: December 12, 2023. Revised: April 24, 2024. Accepted: July 23, 2024.