ON THE PROBLEM OF RESOLUTION OF SINGULARITIES IN POSITIVE CHARACTERISTIC

Transcription

1 July ON THE PROBLEM OF RESOLUTION OF SINGULARITIES IN POSITIVE CHARACTERISTIC (Or: A proof that we are still waiting for) HERWIG HAUSER Introdution. The embedded resolution of singular algebrai varieties of dimension > 3 defined over fields of harateristi p > 0 is still an open problem. The indutive argument whih works in harateristi zero fails for positive harateristi. The main obstrution is the failure of maximal ontat, whih, in turn, manifests in the ourene of wild singularities and kangaroo points at ertain stages of a sequene of blowups. At these points the standard harateristi zero resolution invariant inreases instead of dereasing. The indution breaks down. No remedy has been found yet. In this artile, whih is mostly expository, we will give a detailed disussion of the obstrutions to resolution in positive harateristi. The desription of wild singularities is based on the notion of oblique polynomials. These are homogeneous polynomials showing a speifi behaviour under linear oordinate hanges, whih, in turn, determines them ompletely. Blowing up a wild singularity may ause the appearane of kangaroo points on the exeptional divisor. They represent one of the main problems for establishing the indution in positive harateristi. The proofs and the tehnial details an be found in the original preprint [Ha1], whih is urrently being revised and updated, f. [Ha7]. While addressing us mainly to algebrai geometers with some experiene in resolution matters, we will add in footnotes explanations and referenes for readers that are urious about the reent developments but less familiar with the topi. Setions A and B develop the overall outset of the resolution of singularities, setions C and D then exhibit the speifi problems related to fields of positive harateristi. These setions are written for a general audiene. Starting with setion E, the reader will find more detailed information and preise statements. A. Prelude for the non-expert reader. Before getting into the atual material, let me tell you what is resolution about and why it is important (and, also, why it is so fasinating). Readers aquainted with the subjet may proeed diretly to the next but one setion. A system of polynomial equations in n variables has a zeroset the assoiated algebrai variety X whose struture an be quite ompliated and mysterious. You may think of the real or MSC-2000: 14B05, 14E15, 12D10. Supported within the projet P of the Austrian Siene Fund FWF. The author thanks the members of the Clay Institute for Mathematis at Cambridge and the Researh Institute for Mathematial Siene at Kyoto for their kind hospitality. 1

2 omplex solutions of an equation like 441(x 2 y 2 + y 2 z 2 + x 2 z 2 ) = (1 x 2 y 2 z 2 ) 3. The geometry of varieties shows all kind of loal and global patterns whih are diffiult to guess from the equation. In partiular, there will be singularities. These are the points where X fails to be smooth (i.e., where X is not a manifold). At those points the Impliit Funtion Theorem (IFT) annot be used to ompute the nearby solutions. As a onsequene, it is hard (also for omputers) to desribe orretly the loal shape of the variety at its singular points. Resolution of singularities is a method to understand where singularities ome from, what they look like, and what their internal struture is. The idea is quite simple: When you take a submanifold X of a high dimensional ambient spae M and then onsider the image X of X under the projetion of the ambient spae onto a smaller spae M, you most often reate singularities on X. The Klein bottle is smooth as a submanifold of R 4, but there is no smooth realisation of it in R 3. You neessarily have to aept self-intersetions. Similarly, if you projet a smooth spae urve onto a plane in the diretion of a tangent line at one of its points, the image urve will have singularities. Whih singular varieties an we obtain by suh projetions? The answer is simple: All! Theorem. (Hironaka 1964) Every algebrai variety over C is the image of a manifold under a suitable projetion. Suh a manifold and map an be expliitly onstruted (at least theoretially). For a geometer, this is quite amazing. For an algebraist, this is even more striking, sine it means that it is possible to solve polynomial equations up to the Impliit Funtion Theorem. The appliations of this result are numerous (it would be worth to list all theorems whose proofs rely on resolution). The reason is that, for smooth varieties, a lot of mahinery is available to onstrut invariants and assoiated objets (zeta-funtions, ohomology groups, harateristi lasses, extensions of funtions and differential forms,...). As the projetion map onsists of a sequene of relatively simple maps (so alled blowups), there is a good hane to arry these omputations over to singular varieties. Whih, in turn, is very helpful to understand them better. Resolution is well established over fields of harateristi zero (with nowadays quite aessible proofs), but still unknown in positive harateristi (exept for dimensions up to 3). Why bothering about this? First, beause (almost) everybody expets resolution to be true also in harateristi p. As the harateristi zero ase was already a great piee of work (built on a truly beautiful onatenation of arguments), it is an intriguing hallenge for the algebrai geometry ommunity to find a proof that does not use the assumption of harateristi zero. But there is more to it: Many virtual results in number theory and arithmeti are just waiting to beome true by having at hand resolution in positive harateristi. 1 Again, it would be interesting to produe a list. Another important feature of suh a proof is our understanding of solving equations in harateristi p. If we agree not to aim at one stroke solutions but to simplify the equation step by step (using for instane blowups) until we an see the solution (again, modulo IFT) there appears this deliate matter of understanding loal oordinate hanges in the presene of the Frobenius homomorphism. Phrased in very down to earth terms this means: How do you measure whether a polynomial is, up to oordinate hanges and up to adding p-th 1 De Jong s theory of alterations, valid in arbitrary harateristi but slighty weaker than resolution, already produed a swarm of suh results, f. the next setion. 2

3 power polynomials, lose or far from being a monomial. This is less naive than it may sound: It is an extremely tough question (it has resisted over 50 years), and it lies at the very heart of the resolution of singularities in harateristi p. A meaningful proposal for suh a measure (whih should be ompatible with blowups in a well defined sense) ould break open the wall behind whih we suspet to see a proof of resolution in positive harateristi. The rest would be mainly tehnialities. In the present artile, we will see some of these elementary harateristi p features, and we will make them very expliit. Of ourse it would be nie to have in parallel the oneptual ounterparts of these onstrutions and phenomena, but this would require muh more spae and effort (for both the reader and the writer). As a onsolation, the problems will be so onrete that everybody with a minimum talent in algebra will be tempted to attak them. The more geometrially oriented reader is referred to the survey [FH] for various visualizations of the resolution of surfae singularities. At the end of this paper, we briefly desribe the present state of researh in resolution of singularities in positive harateristi and arbitrary dimension (work of Hironaka, Villamayor, Kawanoue-Matsuki, Włodarzyk). B. Resumé of tehniques and results. This setion will explain the main resolution devies that work independently of the harateristi. The material has beome lassial, with many exellent referenes. After this survey setion we will return to the failure of maximal ontat and the desription of wild singularities and kangaroo points in positive harateristi. By far the most important modifiation of a variety is given by the onept of blowup. Every blowup omes with a enter (a arefully hosen subvariety of our variety), whih is the lous of points where the variety is atually modified. Outside the enter, the variety remains untouhed. The enter itself is replaed by a larger subvariety, whih affets the way how the variety approahes this lous. The hope is that blowups gradually improve the singularities of the varieties until, after possibly many steps, all singularities are eliminated. Whereas this elimination is granted in harateristi zero if one hooses the orret sequene of blowups, the situation is muh more deliate in positive harateristi. The main diffiulty is to measure the omplexity of a singularity by an invariant (usually a lexiographially ordered vetor of integers) in a way suh that after eah blowup (hosen suitably) the invariant has dereased. This is preisely the theme of this artile: The invariant that works well in zero harateristi tends to behave erratially in positive harateristi. Apart from blowups, the main two other harateristi-free tehniques are normalizations and alterations. They will be desribed at the end of this setion. Blowups. We will start with a short introdution to blowups. There are many equivalent ways to define them (see e.g. [EiH, Ha5]). We shall hoose the most geometri and intuitive desription. By an affine variety we shall always understand a subset of affine spae A n K over a field K that is the zeroset X of a bunh of polynomials in n variables. We do not assume that X is irreduible. The oordinate ring K[X] is the quotient of K[x 1,..., x n ] by the ideal I X generated by these polynomials 2. 2 All what follows an be defined for arbitary shemes, the enter being a losed subsheme. See the paragraphs after the examples below for a more oneptual definition of blowups. 3

4 A blowup of X is a new variety X, the blowup or transform of X, together with a morphism π : X X, the blowup map. Any blowup is determined by its enter Z. This is a losed, non-empty subvariety Z of X, usually smooth and inluded in the singular lous Sing(X) of X (the lous of points where X is not smooth; below we define the blowup with respet to an arbitrary ideal). The definition of X and π does not need the embedding X A n, but the explanation is easier if we use it. Moreover we shall assume that we are given a projetion ρ : X Z (again, this is not substantial, but makes things simpler). We should think of X being fibered by transversal setions along Z. If Z is just a point, ρ is the onstant map. If Z is a line, then ρ is typially the restrition to X of a linear orthogonal projetion from A n to Z (having hosen some salar produt on A n ). For any point a of X not in Z there is a unique line in A n, all it l a, passing through a and its projetion point ρ(a) in Z. This is just the seant line through a and ρ(a), and belongs to the spae L(A n ) of all lines in A n. In partiular, the notion of limit line makes sense when a approahes ρ(a) inside the fiber of ρ. The idea of blowups onsists in pulling apart X inside a larger ambient spae. As l is defined on X \ Z and takes values in L(A n ), the graph Γ(l) of l will be an algebrai subset of (X \ Z) L(A n ). It allows to see X \ Z embedded into (X \ Z) L(A n ) via a (a, l a ). We now extend this embedding to the whole of X. The result will no longer be an embedding, but rather a subvariety X of X L(A n ) whih projets onto X. Above points a of Z, there will in general be several points in X, namely all the limiting positions of seants through a. Taking the limit of the seants l a as a X \ Z approahes Z orresponds to adding the boundary points of Γ(l) (X \ Z) L(A n ) when onsidered as a subset of X L(A n ). More preisely, let X be the Zariski-losure of Γ(l) in X L(A n ), i.e., the smallest algebrai subset ontaining Γ(l). Then X will be the blowup of X along Z, and π : X X is the restrition of the projetion X L(A n ) X. Intuitively, we an interpret l(a) as the height of the point on X projeting to a X \ Z. Above points a in Z, this height will in general be multi-valued. All this an be made very preise, and has both algebrai and axiomati interpretations (see the definition after the next examples and the referenes). For the moment, we apply this tehnique to speifi geometri situations. As the blowup map will be an isomorphism over X \ Z (by definition), and sine we have no need to modify the smooth points of X, we shall always hoose the enter Z, as mentioned earlier, inside the singular lous Sing(X) of X. Example 1: Let X be the one in the three-dimensional real affine spae A 3 R of equation x 2 + y 2 = z 2, and let the enter Z be its unique singular point, the origin 0. We laim that the blowup X of X in 0 is the ylinder x 2 + y 2 = 1. This an be heked algebraially, but it is nier to onvine ourselves by a geometri argument. See the lines in A 3 through 0 as elements of projetive spae P 2. Our height-funtion l : X \0 P 2 is defined by assoiating to any point a X \ 0 the line through a and 0 (whih is just a generating line of the one). As a moves on the one straight towards 0, the line l a will always be the same, so the map l is onstant on the lines of X. Clearly, the limiting positions form a irle, and we onlude that X is indeed the ylinder. Example 1 : Take for X the plane urve of equation x 2 = y 2 + y 3, the node. The natural enter to hoose is the singular point 0. The same reasoning as before shows that X is a smooth urve. Also, taking the artesian produt Y of X with a perpendiular axis in A 3 (seeing A 2 as the plane A 2 0 in A 3 ) will pose no problems: the enter Z is now the z-axis, we fiber A 3 by the planes A 2 {t} with t varying in A 1, and get a blowup Y of Y whih is 4

5 the artesian produt of the blowup X of X with the z-axis. Example 2: The usp X of equation x 2 = y 3 is slightly more ompliated to treat. The blowup of X with enter the origin 0 will be a spae urve X in the three-dimensional ambient spae A 2 P 1. As X has just one limit of seants at 0 (the y-axis), there is preisely one point on X sitting above 0 X. Call it a. We have to hek whether X is smooth or singular at a. Unfortunately, this an no longer be done by purely geometri methods, and we have to resort to algebra. Points a on X are of the form (t 3, t 2 ), the respetive seant, taken as an element in P 1, has projetive oordinates (t 3 : t 2 ) = (t : 1), so that the points of X are parametrized by t (t 3, t 2, t). Hene X is smooth at a = 0. The same omputation applied to the sharper usp Y defined by x 2 = y 5 yields for Y the parametrization (t 5, t 2, t 3 ). This shows that Y is still a singular urve. Projet Y to the plane A 2 by forgetting the first omponent. The image urve is the ordinary usp X. By onstrution, X is isomorphi to Y. Therefore another point blowup suffies to resolve Y. There is an algebrai and slightly more general notion of blowup whih is related to an arbitrary ideal N in K[x 1,..., x n ] (now K an be any field). The geometri version above is reovered by taking for N the radial ideal I Z defining Z in A n. Let g 1,..., g k be a system of generators of N, and let Z A n be the ommon zeroset of the g i (whih oinides with the subvariety of A n defined by N). Then the map g : A n \ Z P k 1, a (g 1 (a) :... : g k (a)) is well defined. The Zariski losure Ãn of its graph in A n P k 1 is defined as the blowup of A n with enter N. It is easy to see that Ãn is a variety of dimension n, and isomorphi to the blowup defined geometrially above in ase N is the radial ideal of the subvariety Z of A n. In partiular, Ãn is smooth if Z is smooth. The restrition to Ãn of the first projetion A n P k 1 A n yields the blowup map π : Ãn A n. Embedded resolution. We will have to distinguish between embedded and non-embedded resolution. To explain the differene, let our singular variety X be embedded in some smooth ambient spae W, say, for simpliity, W = A n. Let Z be a subvariety of X (our hosen enter of blowup). It is a general fat that the blowup X of X along Z an be onstruted from the blowup of W along Z. To this end, denote by π : W W the blowup map, and onsider the inverse image X = π 1 (X) of X under π. The variety X is alled the total transform of X. It turns out that X has two omponents. The first is the exeptional divisor E = π 1 (Z) W given by the pull-bak of the enter. It is a hypersurfae in W whih ontrats under π to Z, whereas outside E the map π is an isomorphism onto X \ Z. The seond omponent, say X, is the geometrially interesting objet. It oinides with the blowup of X along Z and is alled the strit transform of X under π. Taking the inverse image π 1 (X \ Z) of X \ Z in W, the Zariski-losure of π 1 (X \ Z) in W gives X. An embedded resolution of X W is a birational proper morphism π : W W so that the total transform X is a variety with at most normal rossings 3. This signifies that the strit transform X of X under π is smooth and transversal to the omponents of the exeptional divisor E in W. 3 Birational morphism: A map given loally by quotients of polynomials induing an isomorphism of a dense open subset onto a dense open subset. Proper: The preimage of ompat sets is ompat. Normal rossings: Loally, the variety is up to isomorphism a union of oordinate subspaes, or, equivalently, an be defined by a monomial ideal. 5

6 In ontrast, a non-embedded resolution of X is just a birational proper morphism ε : X X with X smooth. It does not take into aount the embedding of X but onsiders X as an abstrat variety. We should think of ε as a parametrization of the singular variety X by the smooth variety X. A basi result in birational geometry says that any projetive birational morphism is given as a single blowup of X along a enter Z defined by a possibly very ompliated ideal. In partiular, this holds for any resolution, where now the enter should be supported on Sing(X). Even in the first non-trivial examples it is not lear how to define suh a enter ab initio in order to get via the indued blowup the required resolution. For many appliations one needs embedded resolution. The onept has a variant known as log-resolution of varieties, respetively prinipalization or monomialization of ideals the ideal will be the one defining X in W 4. If a non-embedded resolution ε : X X is given by a sequene of blowups in ertain enters and if we have an embedding X W of X into a smooth variety W, one may take the suessive blowups of W defined by these enters. This yields a birational morphism π : W W of smooth varieties together with an embedding X W. At this stage, X need not meet the exeptional divisor E of π transversally (here, E is defined as the subvariety of W where π is not an isomorphism). But then one an apply further blowups to W until all omponents of the transform of X and E do meet transversally, whih then provides an embedded resolution of X in W. Small dimensions. Let us now turn to resolution of urves, surfaes and three-folds in arbitrary harateristi. The resolution of urves is governed by the fat that all singular points are isolated, so that only point blowups have to be onsidered. One an hoose any of the singular point as enter. The order in whih these are taken does not matter. So the only problem is to show that after finitely many suh blowups the resulting urve is smooth (and, for the embedded resolution, transversal to the exeptional divisor). This is done by defining a loal invariant at eah of the singular points of the urve and showing that, after one blowup, this invariant has dropped at all singular points sitting above the enter. Various invariants for this task have been proposed and work, see the first hapter of Kollár s book desribing thirteen ways how to resolve urve singularities [Ko]. The most frequent invariant for plane urves onsists of two numbers (o, s) (onsidered lexiographially), where o is the order of the Taylor expansion of the defining equation at the point, and s is the first harateristi number (whih we do not define here). For a detailed disussion of this invariant in arbitrary harateristi and the proof that it drops under blowup you may onsult the survey [HR]. Let us next onsider singular surfaes that are embedded as hypersurfaes in a smooth threedimensional ambient spae. The singular lous onsists now of isolated points and (possibly) urves, whih themselves an be singular 5. The isolated points an be taken as the enter of a blowup as before, with the task to exhibit numerially the improvement of the singularities after the blowup. The first ompliation is due to the fat that an isolated singularity may produe under blowup on the transformed surfae a whole urve of singular points. The seond ompliation stems from the urves along whih the surfae is singular. If the urve 4 For a log-resolution one requires that the total transform is in addition a divisor, say a hypersurfae of the smooth ambient spae. This an be ahieved from an embedded resolution by an extra blowup with enter the entire strit transform of the variety. 5 In pratie, one onsiders instead of the singular lous the usually smaller top lous defined earlier as the set of points where the loal order of the defining equation is maximal; the same reasoning applies with slight modifiations. 6

7 is smooth, it an be taken as enter, with the hope of getting an improvement. If it is singular, its singular points are the only reasonable enters, beause blowups whose enter is a singular urve are very diffiult to ontrol. So we start with point blowups. By resolution of urves, finitely many blowups resolve these urves (i.e., make them smooth). On the way, new urves may appear in the singular lous of the surfae. Zariski has shown that they are always smooth. This allows to onlude that after finitely many blowups the singular lous of the surfae onsists of isolated points and smooth urves that, moreover, interset transversally. From that point on we take also urves as enters: Any omponent of the singular lous of the surfae may be hosen (again, the order does not matter, as Zariski showed). It remains to show that the sequene of blowups (whih is geometrially motivated) does indeed resolve all singularities. To this end it suffies to show that the order of the defining equation must drop after finitely many blowups. This problem and the solution to it are known as the theorem of Beppo Levi. Again an invariant that drops after eah blowup has to be defined. There are several proposals. Zariski was able to onstrut one for harateristi zero, and Abhyankar was the first to give a proof of termination in positive harateristi [Za, Ab1]. Hironaka later defined a different and harateristi free invariant based on the Newton polyhedron of the defining equation [Hi4, Ha3]. The onstrution is quite speial and does not seem to apply to higher dimensions. Hauser and Wagner showed, relying on a proposal of Zeillinger, that the nowadays standard harateristi zero invariant of Villamayor, Bierstone-Milman and suessors that works in arbitrary dimension (but may inrease for blowups in positive harateristi) an be modified suitably in ase of surfaes by subtrating a bonus from it. This bonus is a small orretion term whih takes values between 0 and 1 + δ aording to the internal struture of the defining equation, see the setion on the resolution of surfaes. The modified invariant then dereases after eah blowup and thus provides an indution argument [HW]. Quite reently, Cossart, Jannsen and Saito established resolution for surfaes that are not neessarily hypersurfaes over a field by extending Hironaka s onstrution to arbitrary exellent two-dimensional shemes [CJS]. It turns out that all these tehniques atually allow to produe an embedded resolution (sine we are working with the defining equations of the surfae). In ontrast, Lipman s proof of resolution of surfaes via normalization plus blowups yields a non-embedded resolution [Lp1, At] 6. The situation for three-folds is muh more involved. At the moment only non-embedded resolution is established (in arbitrary harateristi). The proof relies vitally on the embedded resolution of surfaes. Abhyankar gave a long proof (more than five hundred pages) that is sattered over several papers and requires that the harateristi of the algebraially losed ground field is > 5. Cutkosky was then able to make this proof muh more transparent and to redue it to less than forty pages [Cu]. In Cutkosky s paper, Abhyankar s work is desribed in great detail, giving all neessary referenes. Cossart and Piltant sueeded to remove the restrition on the harateristi and the algebrai losedness of the ground field. The resulting proof is rather long and hallenging [CP1, CP2], based on ideas of [Co]. Normalization. All the above approahes use in some way or other the modifiation of a variety by blowups. Let us now desribe two alternatives. An important way to improve the singularities of a variety is by means of its normalization. This is an extremely elegant, harateristi independent method to get rid of all omponents 6 The proof selets among all normal varieties proper over the ground field and birational to X one of minimal arithmeti genus, shows that all its singularities are pseudo-rational, and then resolves these by point blowups. 7

8 of the singular lous of odimension one in the variety (e.g., urves in the singular lous of a surfae). One says that the variety beomes regular in odimension one. The onstrution does not look at the embedding. The normalization is defined through the integral losure of rings. Assume that X is an irreduible algebrai subset of affine spae A n over the ground field K, and let R be the oordinate ring of X whose elements are the polynomial funtions on X. The ring R is a finitely generated K-algebra and an integral domain. Let Q be its field of frations (the funtion field of X). Now reall that any morphism f : X X of varieties indues a dual ring homomorphism f : R R between the oordinate rings given by omposition with f. If the morphism f is birational, the map f is injetive and indues an isomorphism of funtion fields Q = Q. Identifying Q with Q, the morphism f an then be read as a ring extension R R Q. This observation suggests to look at overrings of R inside Q that are again finitely generated K-algebras (in order to be the oordinate ring of a variety) and so that the orresponding variety is loser to a smooth variety than X. One answer to this approah is the integral losure R of R in Q. It an be shown that R is a finitely generated K-algebra, and that the extension R R is finite. Therefore R is the oordinate ring of a variety X, and the inlusion R R defines a finite morphism X X, the normalization map. The variety X is normal (its oordinate ring is integrally losed), in partiular, it is regular in odimension one. For urves, this signifies to be smooth (giving a non-embedded resolution), for surfaes we will only have isolated singularities (whih is good for many purposes, but not yet a resolution). It an be shown that iterated ompositions of normalizations and point blowups allow to resolve surfaes. Alterations. The last method that we shall mention in this introdutory part is the notion of alterations introdued by de Jong [dj]. It works in all harateristis, but yields a resolution only up to a finite map. This, however, is suffiient for many appliations [Be]. Let us briefly desribe the idea. Whereas a modifiation of a variety X is a birational proper morphism π : X X yielding an isomorphism of funtion fields, an alteration is a proper, surjetive morphism that indues a finite extension of funtion fields. Geometrially speaking, π is an isomorphism, respetively a finite morphism over a (dense) open subset U of X (generi isomorphism, respetively generially finite morphism). A modifiation is a birational alteration, and an alteration fators into a modifiation followed by a finite map. De Jong shows that any variety (say, over an algebraially losed field) admits an alteration ε : X X with X smooth (and quasi-projetive) [dj, Be, AO]. For the proof by indution one needs a stronger and more preise statement: If S is a losed subvariety of X, the alteration ε an be hosen together with an open immersion i : X Y into a projetive and smooth Y so that the union i(ε 1 (S)) (Y \ X ) forms a normal rossings divisor in Y. The method of proof is opposite to the resolution proofs via blowups: After a preliminary alteration whih allows to assume X to be projetive and normal, the variety X is fibered in urves by onstruting a suitable morphism to a variety P of dimension one less than the dimension of X. This may reate singularities in the fibres whih lie outside the singular lous of X. Using then the theory of semi-stable redution a further alteration together with indution on the dimension of the base spae redues to the ase where the fibres have at most nodal singularities (i.e., are defined loally by xy = 0), and the singular fibres sit only over the points of a normal rossings divisor of P. The situation has then beome so expliit that 8

9 it an be treated by tori methods, yielding finally an alteration ε : X X of X with the required properties. This onludes our summary on resolution. We now turn to the main theme of the artile, the obstrutions to the resolution of singularities in positive harateristi and arbitrary dimension. C. Failure of maximal ontat. There is a onrete reason why resolution is more diffiult in positive harateristi: The behaviour of the singularities under blowup is muh more errati than in harateristi zero. Therefore it is harder to pinpoint and then measure a ontinuous improvement of the singularities yielding eventually to a resolution. In this setion we explain this phenomenon. Some preliminary material is neessary. For the ease of the exposition, we restrit to hypersurfaes X defined by one equation f = 0 in a smooth ambient spae W (e.g. affine spae A n ). Fix a point a of X. Then X is smooth at a if and only if the order of the Taylor expansion of f at a is 1, i.e., if the expansion starts with a linear term. If a is a singular point of X, the order is at least 2. Denote by Sing(X) the set of all singular points of X. This is an algebrai subset, alled the singular lous of X; it is defined by the vanishing of the partial derivatives of f. The omplexity of the singularity of X at a point a Sing(X) is related to the order of vanishing of f at a. Denote this number by ord a X, and by ord(x) the maximal value of ord a X on X. As ord a X is an upper semiontinuous funtion in a and X is a noetherian spae with respet to the Zariski topology, ord(x) is finite and the set top(x) of points of X where ord a X attains its maximum ord(x) is an algebrai subset. This top lous ollets the worst singularities of X. Zariski alls it the equimultiple lous [Za]. The objetive of the resolution proess is to make ord(x) drop by a sequene of blowups in well hosen enters until it beomes 1. Then X will have beome everywhere smooth 7. As blowups are isomorphisms outside the enter, they will not hange the loal order of X there. Sine we are only onerned in a first instane to improve X along top(x) the natural hoie of enter is therefore Z = top(x). The problem is that in general the top lous may itself be singular. Blowing up the smooth ambient spae W in a singular enter reates a new ambient spae W whih now may be singular, and whose singularities an be hard to ontrol. It is then unknown how to measure a possible improvement of the transform X of X in W. Therefore we are onfined to hoose always smooth enters Z. Something nie happens. Fat. Let Z top(x) be a smooth enter, let π : W W be the indued blowup with (strit) transform X of X in W. Let a be a point in Z, and let a be a point in E = π 1 (Z) W mapping under π to a. Then ord a X ord a X. In partiular, we get ord(x ) ord(x) for the maximum value of the loal orders. This says that the omplexity of the singularities of X does at least not get worse. If ord(x ) < ord(x) we an apply indution. If ord(x ) = ord(x) there will be at least one point a E with image a Z and suh that ord a X = ord a X. We all suh points equionstant points (lassially, they are also alled infinitely near points). They are the points where indution annot be diretly applied. Some refined argument is neessary. 7 For an embedded resolution, one has to onsider the total transform of X and try to make it into a normal rossings variety. It is not known how to measure properly the distane of a singularity from being normal rossings. 9

10 One might hope that ord(x) always drops. This is immediately seen to be too optimisti, equality may indeed our. One situation where equality must our is the ase when top(x) is singular. As the enter Z is required to be smooth, Z is then stritly inluded in top(x). Therefore, ord a X remains onstant equal to ord a X for all points a above a top(x) \ Z. Hene ord(x ) = ord(x). Moreover, by the above fat and the upper semiontinuity of the order, it follows that ord a X = ord a X also holds for all points a above a Z. So there is no obvious improvement. To respond to this quandary, one may try to make first Y = top(x) smooth by some auxiliary blowups, in order to take it afterwards as the enter of the next blowup. This fails in two diretions: First, the blowup Y of Y need not oinide with the top lous of the transform X of X. New and even singular omponents may pop up, see [Ha6] for an expliit example. So resolving Y (for instane by indution on the dimension) does not really help to make top(x) smooth (exept for surfaes). Seondly, even if top(x) were smooth and would be taken as enter, it an be shown that ord(x) may not drop under the respetive blowup. From this analysis we learn that the main problem sits in the appearane of equionstant points on the transform X after a blowup. There, the order of X has not dropped, and some refined measure for the improvement of the singularities has to be designed (provided that an improvement as we hope has ourred; this also depends on the orret hoie of the enter, a question whih we will not address here). The next step is to study in more detail the equionstant points, espeially their loation on X. This may help us to understand better how the singularities transform under blowup when the order remains the same. So fix a point a Z in the enter Z top(x) of a given blowup π : W W, and let a E be an equionstant point of X above a. Zariski already observed that there exists, loally in a neighborhood of a in W, a smooth hypersurfae V ontaining a whose transform V under π ontains all equionstant points a above a [Za]. This restrits onsiderably the loation of these points. Zariski desribes quite expliitly all suh hypersurfaes. Now omes the distintion between zero and positive harateristi. In zero harateristi Abhyankar and Hironaka observed 8 that V an be hosen even so that its transform V not only ontains all equionstant points but moreover has itself a transform V ontaining all equionstant points a sitting above a. And this ontinues like this until the order of X drops. It is thus possible to apture the whole sequene of equionstant points above a by one hypersurfae together with its transforms. Suh loal smooth hypersurfaes, whih aompany the resolution proess, are alled hypersurfaes of maximal ontat (and are known as Tshirnhausen transformations in the terminology of Abhyankar). They play the ruial role for the proof of resolution in harateristi zero by allowing now a loal desent in dimension onsidering there a new resolution problem, all it X, in V and its suessive transforms V, V..., see [Ha2]. Formulating this desent properly is not easy but an be done. The resolution of X in V exists by indution on the dimension (this indution tells us also how to hoose the enters). Having resolved X it an be proven that the singularities of X in the original ambient spae W must also have improved (in a preisely defined way). This is the key argument in harateristi zero. 8 Aording to rumors, one breakthrough happened at the end of the fifties on the oasion of a four day visit of Hironaka at Abhyankar s house. 10

11 In positive harateristi, this argument fails drastially: Hypersurfaes of maximal ontat need not exist. There are examples of (e.g. two-dimensional) hypersurfaes with isolated singularities together with a sequene of (point) blowups where any loal smooth hypersurfae V passing through the singularity eventually loses the sequene of equionstant points sitting above the initial point [Na, Ha2]. This prohibits to apply the same desent in dimension as in harateristi zero. Still, for a single blowup, one an hoose, by Zariski s observation, loally at a top(x) a smooth hypersurfae V in W whose transform V ontains the equionstant points a of X in W. The defet is just that this transform V an possibly not be taken again for the subsequent blowups. In this situation, Abhyankar proposed, at least for plane urves, to hange after eah blowup if neessary the hypersurfae. Again one gets a sequene of hypersurfaes, but they will no longer be related as transforms of eah other under blowup. The desent beomes more ompliated. Moreover, there is a priori no anonial hoie for those hypersurfaes. In the next setion we shall desribe this desent in more detail and explain how one an still formulate a resolution problem in smaller dimension. However, its solution is muh harder and has only be ahieved up to now for urves and surfaes. D. Kangaroo phenomenon. Reall that the now lassial resolution invariant in harateristi zero onsists of a vetor of integers whose omponents are orders of ideals in dereasing dimensions. 9 The ideals are the onseutive oeffiient ideals in hypersurfaes of maximal ontat, and the vetor is onsidered with respet to the lexiographi ordering. Two things are then shown: That the lous of points of X where the invariant attains its maximal value is losed, smooth and transversal to the possibly already existing exeptional divisor (stemming from earlier blowups). And, that the invariant drops under blowup when taking as enter this lous of maximal values, as long as the ideals in lower dimension are not resolved yet (in a preise sense). The derease allows to apply indution (the lexiographi order is a well-ordering) and to redue by a finite sequene of blowups to the ase where the invariant attains its minimal possible value. We arrive in this way in the so alled monomial ase, for whih an instant ombinatorial desription of the resolution is known. This program appears in different disguises in many plaes, see e.g. [Hi5, Vi1, BM, EV1, EH, Wł, Ko]. In setion E we will review the harateristi free version of the harateristi zero invariant of an ideal at a point as it was developed in [Ha1, EH]. For this definition, hypersurfaes of maximal ontat (whih need not exist in arbitrary harateristi) have to be replaed by hypersurfaes of weak maximal ontat. These are defined as loal smooth hypersurfaes that maximize the order of the oeffiient ideal of the given ideal (as hypersurfaes of maximal ontat do), but whose transforms, in ontrast, are not required to ontain along a sequene of blowups the points where the order of the original ideal remains onstant. Take then as resolution invariant the lexiographi vetor onsisting of the order of the ideal and of the orders of the iterated oeffiient ideals with respet to suh hypersurfaes. It turns out that the resulting vetor (more preisely, its seond omponent given by the order of the first oeffiient ideal) may inrease in positive harateristi under ertain (permissible) blowups. The first examples of this phenomenon were observed by Abhyankar, Cossart, Moh and Seidenberg [Co, Mo, Se]. The inrease destroys at first glane any kind of indution. 9 For the basis on resolution in harateristi zero, you may onsult [Ha2, Lp2, Ko]. 11

12 Moh sueeded in bounding the maximal inrease, but it was not yet possible to profit from this bound so as to save the indution argument (exept for surfaes). We shall desribe aurately the situations where an inrease of the invariant ours. To make the inrease happen, the variety whih is blown up must have a wild singularity. It is loated at a so alled antelope point of the urrent stage of the sequene of blowups we are onsidering. On the transform of the variety, the inrease of the invariant an then only our at a kangaroo point. 10 The loation of these points and the struture of the singularities is meanwhile well understood and an be explained quite expliitly (f. setion G). Kangaroo points always lie on the new exeptional omponent of the last blowup but never on the transforms of the (old) exeptional omponents passing through the preeding antelope point (see Figure 1). This phenomenon is also known as the ourene of a a translational blowup. a0 a1 antelope a 2 oasis new new old kangaroo a3 new old old Figure 1: The onfiguration of kangaroo, antelope and oasis points. To have a wild singularity at an antelope point preeding a kangaroo point, three onditions must hold: The residues modulo p of the multipliities of the exeptional omponents appearing in the defining equation must satisfy a ertain arithmeti inequality, the order of the oeffiient ideal of the equation must be divisible by the order of the equation, and strong restritions on the (weighted) initial form of the defining equation are imposed (f. the theorem in setion G on kangaroo points). It turns out that the initial form of a wild singularity must be equal (up to multipliation by p-th powers) to an oblique polynomial. Oblique polynomials are haraterized by a very peuliar behaviour under linear oordinate hanges when onsidered up to addition of p-th powers. Fixing the exeptional multipliities and the degree, both subjet to the arithmeti and divisibility ondition, it an be shown that there is preisely one oblique polynomial with these parameters (f. setion I). For surfaes, it is possible to show that the harateristi zero resolution invariant dereases in the long run also in positive harateristi, i.e., that the oasional inreases are ompensated by dereases in the blowups before and after them. A first method for proving this is developed in [Ha1] and will be skethed in setion J below. A seond, more systemati approah introduing the bonus of a singularity will appear in [HW]. It adjusts the harateristi zero invariant in the ritial situations by a small orretion value the bonus so as to ensure a permanent derease of the invariant. We emphasize that there are earlier proofs of resolution of surfaes in arbitrary harateristi by Abhyankar, Lipman and Hironaka using different invariants and arguments [Ab1, Lp1, Hi4]. For three-folds and higher dimensional varities, no omplete indution argument for the embedded resolution seems to be known. With these remarks we onlude the general introdution. From the next setion on, more detailed informations will be given. Aknowledgements. The author is indebted to many people for sharing their ideas and insights with him, among them Heisuke Hironaka, Shreeram Abhyankar, Orlando Villamayor, 10 In [Hi1], kangaroo points run under the name of metastati points. 12

13 Santiago Eninas, Ana Bravo, Gerd Müller, Josef Shiho, Gábor Bodnár, Dale Cutkosky, Edward Bierstone, Pierre Milman, Jaroslav Włodarzyk, Bernard Teissier, Vinent Cossart, Mark Spivakovsky, Hiraku Kawanoue, Kenji Matsuki, Li Li, Daniele Panazzolo, Anne Frühbis-Krüger and János Kollár. We thank Dominique Wagner and Eleonore Faber for a areful reading of the text and several substantial improvements, and Roio Blano for helpful programming support. E. The invariant. We define only the first two omponents of the lassial resolution invariant as these suffie for the phenomena to be desribed here. For an ideal sheaf J on a smooth ambient spae W and a point a W denote by J = J a the stalk of J at a. For onveniene, we denote if appropriate by the same harater J the ideal generated in the ompletion Ô W,a of the loal ring O W,a. 11 For a loal smooth hypersurfae V in W through a, the oeffiient ideal of J in V is defined as the ideal o 1 oeff V J = i=0 (a f,i, f J) o! o i, where o = ord a J is the order of J at a, x = 0 is a loal equation for V and f = a f,i x i is the expansion of f with respet to x, with oeffiients a f,i O V,a. Among the many variants of this definition in the literature, the given one suits best our purposes. More speifiations appear in [EH]. In ase J is a prinipal ideal generated by one polynomial f(x, y) = x o + g(y) in A 1+m with variables x and y = (y m,..., y 1 ), the oeffiient ideal of J with respet to the hypersurfae x = 0 is simply the ideal in A m generated by g (o 1)!. The fatorial is only needed to ensure integer exponents when f has other x-terms. The order of the oeffiient ideal at a depends on the hoie of the hypersurfae V, but remains unhanged under passing to the ompletions of the loal rings. The supremum of these orders over all hoies of loal smooth hypersurfaes V through a is a loal invariant of J at a (i.e., by definition, only depends on the isomorphism lass of the omplete loal ring ÔW,a/J). This supremum is if and only if J is bold regular at a, viz generated by a power of a parameter of ÔW,a [EH]. If the supremum is < and hene a maximum, any hypersurfae V realizing this value is said to have weak maximal ontat with J at a. In harateristi zero, hypersurfaes of maximal ontat have weak maximal ontat [EH]. Moreover, their strit transforms under a permissible blowup W W ontain all equionstant points (= infinitely near points in W ), i.e., those points of the exeptional divisor where the order of the weak transform J of J has remained onstant (reall that this order annot inrease if J has onstant order along the enter). In arbitrary harateristi, the supremum of the orders of the oeffiient ideal oeff V J for varying V an be used to define the seond omponent of the andidate resolution invariant of J at a. If the supremum is and thus J is bold regular, a resolution is already ahieved loally at a, so we disard this ase. We heneforth assume that the supremum is < and an thus be realized by the hoie of a suitable hypersurfae V. After fatoring from the resulting oeffiient ideal a suitable divisor one takes the order of the remaining fator as the seond omponent of the invariant. More expliitly, let D be a given normal rossings divisor 11 You may think here that J is an ideal in a polynomial ring and J is the indued ideal in a formal power series ring generated by the Taylor expansions of the elements of J at a point. 13

14 in W with defining ideal I W (D). We shall assume throughout that oeff V J fators for any hosen V transversal to D (in the sense of normal rossings) into a produt of ideals oeff V J = I V (D V ) I, where I is some ideal in ÔV,a (this assumption is always realized in pratie). Then define the shade of J at a with respet to D as the maximum value shade a J of ord a I over all hoies of V transversal to D. In [Hi1], a similarly defined invariant is onsidered by Hironaka and alled there the residual order of J at a. As usual, questions of well-definedness and upper-semiontinuity have to be taken are of. 12 Along a resolution proess, D will always be supported by the exeptional omponents aumulated so far. It oinides with the seond entry of the ombinatorial handiap of a mobile as defined in [EH]. At the beginning, or whenever ord a J has dropped, D will be empty. If the order of J has remained onstant at a point a above a, the transform D of D is defined as D = D + (ord a (D V ) + shade a J ord a J) Y, where Y denotes the exeptional divisor of the last blowup, and D the strit transform of D. 13 The formula signifies that D onsists of the transform of D together with the new exeptional omponent Y (whih is taken with a suitable multipliity). Note that ord a (D V ) + shade a J = ord a (oeff V J). It follows from the transformation rule of D that, under permissible blowup, the weak transform J of J at an equionstant point a above a has as oeffiient ideal oeff V J in the strit transform V of V an ideal whih fators again into a produt I V (D V ) I, with I the weak transform (I ) of I. Here, it is assumed that the enter Z is ontained in V. This is more deliate to ahieve in positive harateristi, due to the example of Narasimhan where the singular lous of J is not ontained loally in any smooth hypersurfae [Na1, Na2, Mu]. It an, however, be realized by refining the usual stratifiation of the singular lous of J through the loal embedding dimension of this lous. We say that the monomial ase ours when the whole oeffiient ideal has beome an exeptional monomial, say oeff V J = I V (D V ) with I = 1. The shade has then attained its minimal value 0. This ase allows a purely ombinatorial resolution of J (f. [EH]). The ommutativity of the passage to oeffiient ideals with blowups an be subsumed as follows, f. [EH, Ha2]. Given a blowup with enter Z ontained in the loal hypersurfae V of W loally at a and transversal to D, we get for any equionstant a in W above a and I = (I ) a ommutative diagram J oeff V J = I V (D V ) I J oeff V J = I V (D V ) I Here, the situation splits aording to the harateristi: In harateristi zero, hoosing for V a hypersurfae of maximal ontat for J at a, the strit transform V onstitutes again a hypersurfae of maximal ontat for J at a. In partiular, both will have weak maximal 12 Semiontinuity works well if only losed points are onsidered. For arbitrary (i.e., non-losed) points, there appear pathologies whih are desribed and studied by Hironaka [Hi1]. 13 We use here impliitly that V and Z are transversal to D. This is indeed the ase in the resolution proess of an ideal or sheme. 14

15 ontat so that the shades of J and J are well-defined. In addition, shade a J an be omputed from shade a J by looking at the blowup V V with enter Z and the ideals I and I (reall that Z V loally at a). As shade a J = ord a I, shade a J = ord a I and I is the weak transform of I, it follows automatially that shade a J shade a J (it is required here that the order of I is onstant along Z, a property that is ahieved through the insertion of ompanion ideals as suggested by Villamayor, f. [EV2, EH]). This makes the indution and the desent in dimension work. In positive harateristi, it is in general not possible to hoose a loal hypersurfae of maximal ontat for J at a. But a hypersurfae of weak maximal ontat will always exist, by definition. So hoose one, say V. The good news is as already Zariski observed [Za] that the strit transform V of V will ontain all equionstant points a of J in the exeptional divisor Y. The bad news is, as Moh s and Narasimhan s examples show, that V need no longer have weak maximal ontat with J at a. Said differently, V need not maximize the order of the oeffiient ideal of the weak transform J of J at a. One may have to hoose a new hypersurfae U at a to maximize this order. As Moh observed [Mo], there is still worse news, sine the hoie of U may produe a shade of J at a whih is stritly larger than the shade of J at a. This destroys the indution over the lexiographially ordered pair (ord a (J), shade a (J)). At least at first sight! F. Moh s bound. In his paper on loal uniformization, Moh investigates the possible inrease of shade a J at equionstant points a of J in the purely inseparable ase f(x, y) = x pe + y r g(y), with ord(y r g) p e = ord f and e 1 [Mo] 14. Here, V defined by x = x n = 0 denotes a hypersurfae of weak maximal ontat for f at a = 0 in W = A n, p is the harateristi of the (algebraially losed) ground field, and y = (x n 1,..., x 1 ) denote further oordinates so that (x, y) form a omplete parameter system of R = ÔA n,0. Moreover, r N n 1 is a multi-exponent whose entries are the multipliities of the omponents of the divisor D V at 0, and y i = 0 defines an irreduible omponent of D V in V for all i for whih r i > 0. All expressions take plae in the algebra of an étale neighborhood of 0 in A n, so that f and possible oordinate hanges are onsidered as formal power series. The shade of f at 0 with respet to the divisor D defined by y r = 0 is given by shade 0 f = ord 0 g, by the hoie of V. Proposition. (Moh) In the above situation, let (W, a ) (W, a) be a loal blowup with smooth enter Z ontained in the top lous of f and transversal to D. Assume that a is an equionstant point for f at a, i.e., ord a f = ord a f = p e, where f denotes the weak (= strit) transform of f at a. Then shade a f shade a f + p e 1. In ase e = 1, the inequality reads shade a f shade a f + 1, whih is not too bad, but still unpleasant. The short proof of Moh uses a nie trik with derivations, thus eliminating all p-th powers from y r g(y). He then briefly investigates the ase where an inrease of the shade indeed ours, showing that in the next blowup the shade has to drop at least by 1 (if e = 1). This, obviously, does not suffie yet to make indution work. 14 It seems that Abhyankar had already observed this inrease. 15