Lectures on Resolution of Singularities (AM-166) (Annals of Mathematics Studies) - Softcover

Kollar, Janos

Softcover

ISBN 10: 0691129231 ISBN 13: 9780691129235

Publisher: Princeton University Press, 2007

Synopsis

Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, János Kollár provides a comprehensive treatment of the characteristic 0 case. He describes more than a dozen proofs for curves, many based on the original papers of Newton, Riemann, and Noether. Kollár goes back to the original sources and presents them in a modern context. He addresses three methods for surfaces, and gives a self-contained and entirely elementary proof of a strong and functorial resolution in all dimensions. Based on a series of lectures at Princeton University and written in an informal yet lucid style, this book is aimed at readers who are interested in both the historical roots of the modern methods and in a simple and transparent proof of this important theorem.

About this title

About the Author

Janos Kollar is a professor of Mathematics at Princeton University.

Excerpt. İ Reprinted by permission. All rights reserved.

Lectures on Resolution of Singularities

By Jnos Kollr

PRINCETON UNIVERSITY PRESS

Copyright İ 2007 Princeton University Press
All right reserved.
ISBN: 978-0-691-12923-5

Contents

Introduction.....................................................................1Chapter 1. Resolution for Curves.................................................51.1. Newton's method of rotating rulers..........................................51.2. The Riemann surface of an algebraic function................................91.3. The Albanese method using projections.......................................121.4. Normalization using commutative algebra.....................................201.5. Infinitely near singularities...............................................261.6. Embedded resolution, I: Global methods......................................321.7. Birational transforms of plane curves.......................................351.8. Embedded resolution, II: Local methods......................................441.9. Principalization of ideal sheave............................................481.10. Embedded resolution, III: Maximal contact..................................511.11. Hensel's lemma and the Weierstrass preparation theorem.....................521.12. Extensions of K((t)) and algebroid curves..................................581.13. Blowing up 1-dimensional rings.............................................61Chapter 2. Resolution for Surfaces...............................................672.1. Examples of resolutions.....................................................682.2. The minimal resolution......................................................732.3. The Jungian method..........................................................802.4. Cyclic quotient singularities...............................................832.5. The Albanese method using projections.......................................892.6. Resolving double points, char [not equal to] 2..............................972.7. Embedded resolution using Weierstrass' theorem..............................1012.8. Review of multiplicities....................................................110Chapter 3. Strong Resolution in Characteristic Zero..............................1173.1. What is a good resolution algorithm?........................................1193.2. Examples of resolutions.....................................................1263.3. Statement of the main theorems..............................................1343.4. Plan of the proof...........................................................1513.5. Birational transforms and marked ideals.....................................1593.6. The inductive setup of the proof............................................1623.7. Birational transform of derivatives.........................................1673.8. Maximal contact and going down..............................................1703.9. Restriction of derivatives and going up.....................................1723.10. Uniqueness of maximal contact..............................................1783.11. Tuning of ideals...........................................................1833.12. Order reduction for ideals.................................................1863.13. Order reduction for marked ideals..........................................192Bibliography.....................................................................197Index............................................................................203

Chapter One

Resolution for Curves

Resolution of curve singularities is one of the oldest and prettiest topics of algebraic geometry. In all likelihood, it is also completely explored.

In this chapter I have tried to collect all the different ways of resolving singularities of curves. Each of the thirteen sections contains a method, and some of them contain more than one. These come in different forms: solving algebraic equations by power series, normalizing complex manifolds, projecting space curves, blowing up curves contained in smooth surfaces, birationally transforming plane curves, describing field extensions of Laurent series fields and blowing up or normalizing 1-dimensional rings.

By the end of the chapter we see that the methods are all interrelated, and there is only one method to resolve curve singularities. I found, however, that these approaches all present a different viewpoint or technical twist that is worth exploring.

1.1. Newton's method of rotating rulers

Let F(x, y) be a complex polynomial in two variables. We are interested in finding solutions of F = 0 in the form y = [phi](x), where [phi] is some type of function that we are right now unsure about.

Following the classical path of solving algebraic equations, one might start with the case where [phi](x) is a composition of polynomials, rational functions and various mth roots of these. As in the classical case, this will not work if the degree of F is 5 or more in y. One can also try to look for power series solutions, but simple examples show that we have to work with power series with fractional exponents. The equation [y.sup.m] = x + [x.sup.2] has no power series solutions for m [greater than or equal to] 2, but it has fractional power series solutions for any [[epsilon].sup.m] = 1 given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a more interesting example, [y.sup.m] - [y.sup.n] + x = 0 for m > n also has a fractional power series solution

y = [[summation].sub.i]greater than or equal to]1] [a.sub.i][x.sup.i/n],

where [a.sub.1] = 1 and the other [a.sub.i] are defined recursively by

n [a.sub.s] = coefficient of [x.sup.(s+n-1)/n] in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

After many more examples, we are led to look for solutions of the form

y = [[infinity].summation over (i=0)] [c.sub.i][x.sup.i/M],

where M is a natural number whose dependence on deg F we leave open for now. These series, though introduced by Newton, are called Puiseux series. We encounter them later several times.

Theorem 1.1 (Newton, 1676). Let F(x, y) be a complex polynomial or power series in two variables. Assume that F(0, 0) = 0 and that [y.sup.n] appears in F(x, y) with a nonzero coefficient for some n. Then F(x, y) = 0 has a Puiseux series solution of the form

y = [[infinity].summation over (i=1)] [c.sub.i][x.sup.i/N]

for some integer N.

Remark 1.2. (1) The original proof is in a letter of Newton to Oldenburg dated October 24, 1676. Two accessible sources are [New60, pp.126-127] and [BK81, pp.372-375].

(2) Our construction gives only a formal Puiseux series; that is, we do not prove that it converges for [absolute value of x] sufficiently small. Nonetheless, if F is a polynomial or a power series that converges in some neighborhood of the origin, then any Puiseux series solution converges in some (possibly smaller) neighborhood of the origin. This is easiest to establish using the method of Riemann, to be discussed in Section 1.2.

(3) By looking at the proof we see that we get n different solutions (when counted with multiplicity).

The proof of Newton starts with a graphical representation of the "lowest order" monomials occurring in F. This is now called the Newton polygon.

Definition 1.3 (Newton polygon). Let F = [summation][a.sub.ij][x.sup.i][y.sup.j] be a polynomial or power series in two variables. The Newton polygon of f (in the chosen coordinates x and y) is obtained as follows.

In a coordinate plane, we mark the point (i, j) with a big dot if [a.sub.ij] [not equal to] 0. Any other monomial [x.sup.i'] [y.sup.j'] with i' [greater than or equal to] i, j' [greater than or equal to] j will not be of "lowest order" in any sense, so we also mark these. (In the figures these markings are invisible, since I do not want to spend time marking infinitely many uninteresting points.)

The Newton polygon is the boundary of the convex hull of the resulting infinite set of marked points.

Assume now that F contains a nonzero term [a.sub.0n][y.sup.n] and n is the smallest possible. This means that the Newton polygon has a corner on the y-axis at the point (0, n). Look at the nonvertical edge of the Newton polygon starting at (0, n). Let us call this the leading edge of the Newton polygon. (As Newton explains it, we put a vertical ruler through (0, n) and rotate it till it hits another marked point-hence, the name of the method.)

1.4 (Proof of (1.1)). We construct the Newton polygon of F and concentrate on its leading edge.

If the leading edge is horizontal, then there are no marked points below the j = n line, and hence, [y.sup.n] divides F and y = 0 is a solution.

Otherwise, the extension of the leading edge hits the x-axis at a point which we write as nu/v where u, v are relatively prime. The leading edge is a segment on the line (v/u)i + j = n. In the diagram below the leading edge hits the x-axis at 7/2, so u = 1 and v = 2.

We use induction on the leading edge, more precisely, on its starting point (0, n) and on its steepness v/u.

Our aim is to make a coordinate change and to obtain another polynomial or power series [F.sub.1]([x.sub.1], [y.sub.1]) with leading edge starting at (0, [n.sub.1]) and steepness [v.sub.1]/[u.sub.1] such that

either [n.sub.1] < n,

or [n.sub.1] = n, and [v.sub.1]/[u.sub.1] < v/u.

Moreover, we can write down a Puiseux series solution of F(x, y) = 0 from a Puiseux series solution of [F.sub.1]([x.sub.1], [y.sub.1]).

Then we repeat the procedure. The first case can occur at most n-times, so eventually the second case happens all the time. We then construct a Puiseux series solution from this infinite sequence of coordinate transformations.

In order to distinguish the two cases, we consider the terms in F that lie on the leading edge

f(x, y) := [[summation].sub.(v/u)i+j=n] [a.sun.ij] [x.sup.i] [y.sup.j],

and we think of this as the "lowest terms" of F. If [a.sub.ij][x.sup.i][y.sup.j] is a nonzero term in F(x, y), then (v/u)i+j [greater than or equal to] n, so f(x, y) indeed consists of the lowest degree terms in F if we declare that deg x = v/u and deg y = 1.

In the above example, f(x, y) = [y.sup.7] + [y.sup.5] x + [y.sup.3] [x.sup.2].

Note that (v/u)i + j = n has an integer solution only if v|n - j; thus we obtain the following.

Claim 1.4.1. We can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In particular, if v [not equal to] 1, then f(1, y) does not contain the term [y.sup.n-1] and so f(1, y) is not an nth power of a linear form.

We distinguish the two cases based on how f(1, y) factors.

Case 1. f(1, y) is not an nth power.

Let [alpha] be a root of f(1, y) with multiplicity [n.sub.1] < n. Then we make the substitutions

x := [x.sup.v.sub.1], y = [y.sub.1][x.sup.u.sub.1] + [alpha][x.sup.u.sub.1].

Note that if [a.sub.ij][x.sup.i][y.sup.j] is a nonzero term in F(x, y), then (v/u)i+j [greater than or equal to] n; thus

[a.sub.ij][x.sup.i][y.sup.j] = [a.sub.ij][x.sup.vi.sub+uj.sub.1] [([y.sub.1 + [alpha]).sup.j]

and vi + uj [greater than or equal to] nu with equality only if (v/u)i + j = n. Thus F]ILT([x.sup.v.sub.1], [y.sub.1][x.sup.u.sub.1] + [alpha][x.sup.usub.1]) is divisible by [x.sup.nu.sub.1], and we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Note furthermore that

[F.sub.1](0, [y.sub.1]) = f(1, [y.sub.1] + [alpha]),

and so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] appears in [F.sub.1] with nonzero coefficient.

Furthermore, any Puiseux series solution [y.sub.1] = [phi]([x.sub.1]) of [F.sub.1] = 0 gives a Puiseux series solution

y = [phi]([x.sup.1/v])[x.sup.u/v] + [alpha][x.sup.u/v]

of F = 0.

Case 2. f(1, y) is an nth power.

By (1.4.1) this can happen only for v = 1. Write f(1, y) = c[(y - [alpha]).sup.n], and make a coordinate change

x = [x.sub.1], y = [y.sub.1] + [alpha][x.sup.u.sub.1].

Under this transformation, [x.sup.i][y.sup.j] becomes a sum of monomials [x.sup.i'.sub.1] [y.sup.j'.sub.1] , where (1/u)i' + j' = (1/u)i + j. Thus we do not get any new terms below the leading edge of f, and we kill every monomial on the leading edge save [y.sup.n], which is now [y.sup.n.sub.1].

Hence [n.sub.1] = n, but the leading edge of the Newton polygon of

[F.sub.1]([x.sub.1], [y.sub.1]) := F([x.sub.1], [y.sub.1] + [alpha][x.sup.u.sub.1])

is less steep than the leading edge of the Newton polygon of F(x, y).

Next we repeat the procedure with [F.sub.1]([x.sub.1], [y.sub.1]) to get [F.sub.2]([x.sub.2], [y.sub.2]) and so on.

As we noted, the only remaining question is, what happens when the second case happens infinitely often. This means that we have an infinite sequence of coordinate changes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here [u.sub.s+1] < [u.sub.s+2] < ...; thus we can view this sequence as converging to a single power series substitution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (invertible power series), giving the power series solution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

1.2. The Riemann surface of an algebraic function

The resolution of singularities of analytic curves is due to Riemann. When he constructs the Riemann surface of a function, he goes directly to the smooth Riemann surface, bypassing the singular model; see [Rie90, pp.39-41]. His method is essentially the one given below.

In more contemporary terminology, here is the result.

Theorem 1.5 (Riemann, 1851). Let F(x, y) be an irreducible complex polynomial and C := (F(x, y) = 0) [subset] [C.sup.2] the corresponding complex curve. Then there is a 1-dimensional complex manifold [bar.C] and a proper holomorphic map

[sigma] : [bar.C] [right arrow] C,

which is a biholomorphism except at finitely many points.

Proof. Since F is irreducible, F and [partial derivative]F/[partial derivative]y have only finitely many points [summation] [subset] ITLITL in common. By the implicit function theorem, the first coordinate projection [pi] : ITLITL [right arrow] ITLITL is a local analytic biholomorphism on ITLITL \ [summation].

We start by constructing a resolution for a small neighborhood of a point p [member of] [summation]. For notational convenience assume that p = 0, the origin.

Let [B.sub.[epsilon]] [subset] [C.sup.2] denote the ball of radius [epsilon] around the origin. By choosing [epsilon] small enough, we may assume that ITLITL [intersection] (y = 0) [intersection] [B.sub.[epsilon]] = {0}.

Next, by by choosing [eta] small enough, we can assume that

[pi] : ITLITL [intersection] [B.sub.[epsilon]] [intersection] [[pi].sup.-1]([[DELTA].sub.[eta]]) [right arrow] [[DELTA].sub.[eta]]

is proper and a local analytic biholomorphism except at the origin, where [[DELTA].sub.[eta]] [subset] C is the disc of radius [eta]. Set

[ITLITL.sub.[eta]] := ITLITL [intersection] [B.sub.[epsilon]] [intersection] [[pi].sup.-1]([[DELTA].sub.[eta]]) and [C.sup.*.sub.[eta]] := [C.sub.[eta]] \ {0}.

We thus conclude that

[pi] : [C.sup.*.sub.[eta]] [right arrow] [[DELTA].sup.*.sub.[eta]] is a covering map.

The fundamental group of [[DELTA].sup.*.sub.[eta]] is Z; thus for every m, the punctured disc [[DELTA].sup.*.sub.[eta]] has a unique connected covering of degree m, namely,

[[[rho].sub.m] : [[DELTA].sup.*.sub.1] [right arrow] [[DELTA].sup.*.sub.[eta]] given by z [??] [eta][z.sup.m].

Let [ITLITL.sup.*.sub.[eta],i] [subset] [ITLITL.sup.*.sub.[eta]] be any connected component and [m.sub.i] the degree of the covering [pi] : [C.sup.*.sub.[eta],I] [right arrow] [[DELTA].sup.*.sub.[eta]]. We thus have an isomorphism of coverings

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

More precisely, topology tells us only that [[sigma].sup.*.sub.i] is a homeomorphism. However, the maps [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [pi] are local analytic biholomorphisms; thus we can assert that [[sigma].sup.*.sub.i] is a homeomorphism that is also a local analytic biholomorphism, and hence a global analytic biholomorphism.

The image of [[sigma].sup.*.sub.i] lands in the ball [B.sub.[epsilon]], and hence the coordinate functions of [[sigma].sup.*.sub.i] are analytic and bounded on [[DELTA].sup.*.sub.1]. Thus by the Riemann extension theorem, [[sigma].sup.*.sub.i] extends to a proper analytic map

[[sigma].sub.i] : [[DELTA].sub.1] [right arrow] [ITLITL.sub.[eta]].

Doing this for every connected component [ITLITL.sup.*.sub.[eta],i] : i [member of] I, we obtain a proper analytic map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an isomorphism, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes disjoint union.

This proves the local resolution for complex algebraic plane curves.

To move to the global case, observe that [summation] [subset] ITLITL is a discrete subset. Thus for each [p.sub.i] [member of] [summation] we can choose disjoint open neighborhoods [p.sub.i] [member of] [C.sub.i] [subset] ITLITL. By further shrinking [C.sub.i], we have proper analytic maps [[sigma].sub.i] : [[bar.C].sub.i] [right arrow] [C.sub.i], where [[bar.ITL.sub.i] is a disjoint union of open discs and [[sigma].sub.i] is invertible outside the singular point [p.sub.i] [member of] [C.sub.i].

We can thus patch together the "big" chart ITLITL \ [summation] with the local resolutions [[bar.ITL.sub.i] to get a global resolution [bar.ITLITL].

1.6 (Puiseux expansion). The resolution of the local branches

[[sigma].sub.i] : [[DELTA].sub.1] [right arrow] [C.sub.[eta],i]

is given by a power series on [[DELTA].sub.1], and the local coordinate on [[DELTA].sub.1] can be interpreted as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus we obtain that each local branch [C.sub.[eta],i] has a parametrization by a convergent Puiseux series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 1.7. There is lot more to Puiseux expansions than the above existence theorems.

(Continues...)

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Edition details

• Publisher: Princeton University Press
• Publication date: 2007
• Language: English
• ISBN 10: 0691129231
• ISBN 13: 9780691129235
• Binding: Paperback
• Number of pages: 216