# MA 206 notes: introduction to resolution of singularities

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1 MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31

2 Resolution of singularities Let k be a field and X a reduced variety over k. The set of regular points X reg of X is Zariski-dense. (Read Liu 4.3, 6.1, 6.2) Definition A resolution of singularities of a reduced variety X is a proper birational morphism f : X X, where X is regular and irreducible, and f restricts to an isomorphism f 1 (X reg ) X reg. (We ll assume X geometrically reduced from now on.) Theorem (Hironaka s Main Theorem 1) Let X be a reduced variety over k with Char(k) = 0. Then there is a projective resolution of singularities X X. We ll mostly assume Char(k) = 0 from now on. Abramovich Introduction to resolution of singularities 2 / 31

3 Method: blowing up and embedded resolution The main tool: if X Y closed with Y regular, find Z X closed and regular with blowing up Ỹ Y such that the strict transform X Ỹ gets better. Theorem (Embedded resolution) Suppose X Y is a closed subvariety of a regular variety Y. There is a sequence of blowings up Y n Y n 1 Y 0 = Y, with regular centers Z i Y i and strict transforms X i Y i, such that Z i does not contain any irreducible component of X i and such that X n is regular. Abramovich Introduction to resolution of singularities 3 / 31

4 From embedded resolution to resolution How to get resolution in general? If X is embedded then you get resolution. In general can blow up so that X is embedded: Chow s 1 Lemma. Best option: X is always locally embedded. We ll make sure the procedure on X is independent of embedding. 1 pronounced Zhou s Abramovich Introduction to resolution of singularities 4 / 31

5 Normal crossings To go further one needs to describe a desirable property of the exceptional divisor E i and its interaction with the center Z i. Definition We say that a closed subset E Y of a regular variety Y is a simple normal crossings divisor if in its decomposition E = E j into irreducible components, each component E j is regular, and these components intersect transversally: locally at a point p E there are local parameters x 1,..., x m such that E is the zero locus of a reduced monomial x 1 x k. We further say that E and a regular subvariety Z have normal crossings if such coordinates can be chosen so that Z = V (x j1,..., x jl ) is the zero set of a subset of these coordinates. Abramovich Introduction to resolution of singularities 5 / 31

6 When the set of coordinates x j1,..., x jl is disjoint from x 1,..., x k the strata of E meet Z transversely, but the definition above allows quite a bit more flexibility. This definition works well with blowing up: If E is a simple normal crossings divisor, E and Z have normal crossings, f : Y Y is the blowing up of the regular center Z with exceptional divisor E Z, and E = f 1 E E Z then E is a simple normal crossings divisor. Abramovich Introduction to resolution of singularities 6 / 31

7 Principalization Theorem (Principalization) Let Y be a regular variety and I an ideal sheaf. There is a sequence of blowings up Y n Y n 1 Y 0 = Y, regular subvarieties Z i Y i, i = 0,..., n 1 and simple normal crossings divisors E i Y i, i = 1,..., n such that f i : Y i+1 Y i is the blowing up of Z i for i = 0,..., n 1, E i and Z i have normal crossings for i = 1,..., n 1, IO Yi vanishes on Z i for i = 0,..., n 1, E i+1 is the union of f 1 i E i with the exceptional locus of f i for i = 0,..., n 1 and such that the resulting ideal sheaf I n = IO Yn is an invertible ideal with zero set V (I n ) supported in E n. We will later require a possibly nonempty E 0 Y 0. Abramovich Introduction to resolution of singularities 7 / 31

8 Principalization implies embedded resolution In local coordinates x 1,..., x m on Y n as above, this means that I n = (x a 1 1 x a k m ) is locally principal and monomial, hence the name principalization. The condition that Z i have normal crossings with E i guarantees that E i+1 is a simple normal crossings divisor. Quoting Kollár (p. 137), principalization implies embedded resolution seemingly by accident : suppose for simplicity that X is irreducible, and let the ideal of X Y be I. Since I n is the ideal of a divisor supported in the exceptional locus, at some point in the sequence the center Z i must contain the strict transform X i of X. Since I vanishes on Z i, it follows that Z i coincides with X i at least near X i. In particular X i is regular! Abramovich Introduction to resolution of singularities 8 / 31

9 Are we working too hard? Principalization seems to require too much for resolution: why should we care about exceptional divisors which lie outside X? Are we trying too hard? In the example of the cuspidal curve X Y = A 2 above, the single blowing up Y 1 Y at the origin does not suffice for principalization: the resulting equation x 2 (z 2 x) = 0 with exceptional {x = 0} is not monomial. One needs no less that three more blowings up! Abramovich Introduction to resolution of singularities 9 / 31

10 The cusp example again I ll describe just one key affine patch of each: Blowing up x = z = 0 one gets, in one affine patch where x = zw, the equation z 3 w 2 (z w) = 0, strict transform X 2 = {z = w} and exceptional {wz = 0}. Blowing up z = w = 0 one gets, in one affine patch where z = wv, the equation v 3 w 6 (v 1) = 0. The exceptional in this patch is {wv = 0}, with one component (the old {z = 0}) appearing only in the other patch. The strict transform is X 3 = {v = 1}. In the open set {v 0} we blow up {v = 1}. This actually does nothing, except turning the function u = v 1 into a monomial along v = 1, so the equation v 3 w 6 (v 1) = 0 at these points can be written as (u + 1) 3 w 6 u = 0, which in this patch is equivalent to w 6 u = 0, a monomial in the exceptional parameters u, w. Abramovich Introduction to resolution of singularities 10 / 31

11 The cusp example: discussion The fact that we could blow up {v = 1} means that X 3 is regular, giving rather late evidence that we obtained resolution of singularities for X. These redundant steps add to the sense that this method works by accident. It turns out that principalization itself is quite useful in the study of singularities. Also the fact that it provides the prize of resolution is seen as sufficient justification. Accident or not, we will continue to pursue principalization. Abramovich Introduction to resolution of singularities 11 / 31

12 Order Finally, principalization of an ideal is proven by way of order reduction. The order ord p (I) of an ideal I at a point p of a regular variety Y is the maximum integer d such that m d p I; here m p is the maximal ideal of p. It tells us how many times every element of I p vanishes at p. In particular we have ord p (I) 1 precisely if I vanishes at p. We write maxord(i) = max{ord p (I) p X }. For instance we have maxord(i) = 0 if and only if I is the unit ideal, which vanishes nowhere. Another exceptional case is maxord(i) = which happens if I vanishes on a whole component of Y. Abramovich Introduction to resolution of singularities 12 / 31

13 Order subsets and admissibility Given an integer a, we write V (I, a) for the locus of points p where ord p (I) a. A regular closed subvariety Z Y is said to be (I, a)-admissible if and only if Z V (I, a), in other words, the order of I at every point of Z is at least a. Admissibility is related to blowings up: if maxord(i) = a, and if Y Y is the blowing up of an (I, a)-admissible Z Y, with exceptional divisor E having ideal I E, then IO Y = (I E ) a I, with maxord(i ) a. In other words, orders do not grow in admissible blowings up Abramovich Introduction to resolution of singularities 13 / 31

14 Order reduction Order reduction is the following statement: Theorem (Order reduction) Let Y be a regular variety, E 0 Y a simple normal crossings divisor, and I an ideal sheaf, with maxord(i) = a. There is a sequence of (I, a)-admissible blowings up Y n Y n 1 Y 0 = Y, with regular centers Z i Y i having normal crossings with E i such that IO Xn = I n I nwith I n an invertible ideal supported on E n and such that maxord(i n) < a. Abramovich Introduction to resolution of singularities 14 / 31

15 Order reduction implies principalization Order reduction implies principalization simply by induction on the maximal order maxord(i) = a: once maxord(i n) = 0 we have IO Xn = I n so only the exceptional part remains, which is supported on a simple normal crossings divisor by induction. Hironaka himself used the Hilbert Samuel function, an invariant much more refined than the order. It is a surprising phenomenon that resolution becomes easier to explain when one uses just the order, thus less information, see [Encinas-Villamayor]. It remains to prove order reduction. Abramovich Introduction to resolution of singularities 15 / 31

16 Tangent and cotangent sheaves Consider a separated morphism Y S, the scheme YY 2 := Y S Y, and the diagonal Y YS 2. It is a closed subscheme isomorphic to Y with ideal I. Definition The relative cotangent sheaf Ω 1 Y /S = I /I 2, naturally an O Y -module. Also known as sheaf of relative Kähler differentials. The sheaf T Y /S = Hom Y (Ω 1 Y /S, O Y ) also denoted Θ Y /S is the tangent sheaf. The universal O Y -derivation d : O Y Ω 1 Y /S relative to S is defined by sending a function f to π1 f π 2 f mod I2. If Y S is a smooth morphism then these are locally free sheaves, just as you learned in Manifolds. We ll mostly work with S = Spec k and drop it from notation, but not always. Abramovich Introduction to resolution of singularities 16 / 31

17 Differential operators: characteristic 0 naïvely Sections of T Y /S are by definition derivations O Y O Y which are zero on O S. Let us start by a naïve definition. Definition The sheaf of rings generated over O Y by the operators in T Y is the sheaf of differential operators D Y. As a sheaf of O Y modules it looks locally like the symmetric algebra Sym (T Y ) = n 0 Sym n (T Y ), but its ring structure is very different, as D Y is non-commutative. Still for any integer a there is a subsheaf D a Y D Y of differential operators of order a, those sections which can be written in terms of monomials of order at most a in sections of T Y. As a special case, one always has a splitting D 1 Y = O Y T Y, the projection D 1 Y O Y given by applying (1). Abramovich Introduction to resolution of singularities 17 / 31

18 Differential operators in general Here is a better way to define things, which detects p-th powers in characteristic p: Definition The sheaf of principal parts of order a of Y is defined as PP a Y = O Y Y /I a+1 This is a sheaf of O Y -modules via either projection; its fiber at p Y describes functions on Y up to order a at p. Its dual sheaf is D a := (PPY a ), which in characteristic 0 admits the concrete description given earlier. The natural projection PP a Y PPa 1 Y inclusion D a 1 Y D Y = a D a Y. D a Y gives rise to an, and one defines in general This is nice enough, but the fact that in positive characteristics sections of D Y are not written as polynomials in sections of T Y is the source of much trouble. Abramovich Introduction to resolution of singularities 18 / 31

19 Derivatives and order Definition Let I be an ideal sheaf on Y and y Y a point. Write D a Y I for the ideal generated by elements (f ), where an operator in D a Y and f a section of I. We have the following characterization: ord y (I) = min{a : (D a I) p = O Y,p }. In other words, the order of I at y is the minimum order of a differential operator such that for some f I y the element (f ) does not vanish at y. Abramovich Introduction to resolution of singularities 19 / 31

20 The order-a scheme We define MC(I, a) := D a 1 I In these terms, the set V (I, a) can be promoted to a scheme, the zero locus of an ideal: V (I, a) := V (MC(I, a)). 2 This is not too surprising in characteristic 0 since we all learned calculus, but it may seem strange in characteristic p > 0. For instance, the order of (x p ) is p, since there is always an operator of order p such that (x p ) = 1. In characteristic 0 we can write = 1 ( ) p, p! x but in characteristic p we have no such expression! 2 This is the right scheme structure, as it satisfies an appropriate universal property. Abramovich Introduction to resolution of singularities 20 / 31

21 Maximal contact Definition Let I be an ideal of maximal order a. A maximal contact hypersurface for (I, a) at p is a hypersurface H regular at p, such that, in some neighborhood Y 0 of p we have H V (I, a) = V (MC(I, a)), namely H contains the scheme of points where I has order a. Proposition In characteristic 0, a maximal contact hypersurface for (I, a) at p exists. Abramovich Introduction to resolution of singularities 21 / 31

22 Existence Proposition In characteristic 0, a maximal contact hypersurface for (I, a) at p exists. Since I has maximal order a, we have D a (I) = (1). Consider the ideal MC(I, a) = D a 1 (I). Since we are in characteristic 0, it must contain an antiderivative of 1, so ord p (MC(I, a)) 1. Any local section x of MC(I, a) with order 1 gives a maximal contact hypersurface {x = 0} at p. Abramovich Introduction to resolution of singularities 22 / 31

23 Classic example Suppose I = (f ) where f (x, y) = y a + g 1 (x)y a g a 1 (x)y + g a (x). Then ord (0,0) (f ) = a exactly when ord 0 (g i ) i for all i. In characteristic 0 we may replace y by y + g 1 (x)/a, so we may assume g 1 (x) = 0. In this case a 1 f / y a 1 = a! y, so {y = 0} is a maximal contact hypersurface at (0, 0). Abramovich Introduction to resolution of singularities 23 / 31

24 Maximal contact is local The definition I gave is pointwise. It is easy to see that if H is a maximal contact hypersurface for (I, a) at p then the same holds at any nearby p, so the concept is local. Unfortunately it is also not hard to cook up examples where there is no global maximal contact hypersurface which works everywhere. We will have to tackle this problem. Positive characteristics: Alas, there are fairly simple examples in charactersitic p > 0 where maximal contact hypersurfaces do not exist [Narasimhan]. The whole discussion from here on simply does not work in characteristic > 0. Abramovich Introduction to resolution of singularities 24 / 31

25 What to restrict to H? Consider I = (f ) with f (x, y) = y a + g 2 (x)y a g a 1 (x)y + g a (x), and ord g i i so maxord(i) = a. In order to reduce the order of I we need to reduce the order of at least one g i so that ord g i < i. So the information necessary needs to involve all these g i. The right generalization is given by the collection of ideals D i (I), i a 1, with maximal order a i, and their restriction to H. Abramovich Introduction to resolution of singularities 25 / 31

26 Dimension reduction This is central and requires work: Proposition Any sequence of (I, a)-admissible blowings up has centers lying in H and its successive strict transforms. The resulting sequence of blowings up on H is ((D i Y I) H, a i)-admissible for every i < a. Conversely, every sequence of blowings up on H which is ((D i Y I) H, a i)-admissible for every i < a gives rise, by blowing up the same centers on Y, to a sequence of (I, a)-admissible blowings up. Abramovich Introduction to resolution of singularities 26 / 31

27 The W lodarczyk ideal Definition (Kollár) Giving D i Y I with i a 1 weight a i, define W (I, a) = D i j Y I (a ij )=a! to be the ideal generated by terms of total weight a!. Order reduction of the collection of ideals is equivalent to that of the W lodarczyk ideal, hence Corollary ((13) Kollár, Corollary 3.85) A sequence of blowings up is an order reduction for (I, a) if and only if it is an order reduction for (W (I, a) H, a!). Traditionally one uses a coefficient ideal instead. j Abramovich Introduction to resolution of singularities 27 / 31

28 Separation of exceptional loci This does not quite work with E 0, which is necessary. In the example of a cuspidal curve above, the ideal I = x 2 (z 2 x) is of the form I 2 E I. The unique maximal contact hypersurface for (I, 1) is precisely X, the vanishing locus of I, but since it is tangent to E it does not have normal crossings with E. One treats this is via a trick: one separates the relevant part of the ideal I from the monomial part I E by applying a suitable principalization for an ideal of the form I α E + I β describing the intersection of their loci. Abramovich Introduction to resolution of singularities 28 / 31

29 Patching Hypersurfaces of maximal contact are not unique and do not patch. However we have Proposition ((4), (W lodarczyk) Let H 1, H 2 Y be two local maximal contact hypersurfaces at p V (I, a). Then, after replacing Y by an étale neighborhood of p, there is an automorphism φ of Y fixing p, V (I, a), stabilizing W (I, a), and sending H 1 to H 2. Abramovich Introduction to resolution of singularities 29 / 31

30 A sketch of the algorithm Let us summarize how one functorially reduces the order of a nonzero ideal I of maximal order a > 0 on a regular variety Y. If dim(y ) = 0 there is nothing to prove, since I is trivial hence of order 0. We assume proven order reduction in dimension < dim(y ). We cover V (I, a) with open patches U possessing maximal contact hypersurfaces H U. The W lodarczyk ideal W (I, a) HU has order a!. If this order is infinite, it means that I U = I a H U, we simply blow up H U and automatically the order of I is reduced on U. Otherwise we can inductively reduce the order of this ideal by a functorial sequence of transformations H k H until the order drops below a!. By Corollary 13 these provide a local order reduction for (I, a) which patches together to a functorial order reduction by Proposition 4. Abramovich Introduction to resolution of singularities 30 / 31

31 The re-embedding principle Proposition (The re-embedding principle) Suppose I is an ideal on a regular variety Y. Consider the embedding Y Y 1 := Y A 1 sending y (y, 0). Let I 1 = IO Y1 + (z), where z is the coordinate on A 1. Then the principalization described above of I 1 on Y 1 is obtained by taking the principalization of I on Y and blowing up the same centers, embedded in Y 1 and is transforms. Abramovich Introduction to resolution of singularities 31 / 31

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