Resolution of Singularities


[1] Introduction

The Problem of Resolutions of Singularities is well--known among mathematicians. Briefly the simplest form is the following:

Basic Problem of Resolution:

Given a finitely generated function field k(x_1,.....,x_n) over $k$. Does there exist a nonsingular projective model for it?

This existence type question is very hard to answer in general. Usually we may begin with some singular projective model $X$ for it. Namely we take some field generators $1,x_1,....,x_n$ or using $1$ to homogenize the defining equations for $x_1,...., x_n$, thus we get a projective model. Note that the singular loci, the collection of all singular points, form a lower dimensional subvariety. This fact can be proved easily by the Jacobian criterion. In other words almost all points are nonsingular. Now the problem may be reformulated as

Problem of Resolution of Singularities:

Assume that $k$ is algebraically closed. Does there exist a non--singular projective model $Y$ and a proper map $f:Y >--> X$ such that $f$ is an isomorphism over some open dense subset $U$ of $X$?

The map $f$ can be understood as an introduction of some hidden parameters at the singularities of $X$. The map $f$ can also be understood as an ``anti--projection''.

In general it is an art to pick up the center of blow--ups.

[2] A Brief History

For curves over the complex numbers $C$. Kronecker, Max Noether and others have solved the Problem of Resolution of Singularities.

For surfaces over C, the first correct solution to Problem of Resolution of Singularities was due to R.Walker in 1936.

Great mathematician O.~Zariski established a profound way of attacking this problem. Let us introduce his concept of uniformization. As is well--known, a Riemann Surface is nothing but the collection of all valuations, with certain nature topology, of a one--dimensional function field. Zariski generalized the above concept and defined ``Zariski--Riemann Surface'' of any function field as the collection of all valuations with a natural topology. Given any projective model $X$ of a function field $K$. Then the Zariski--Riemann Surface of $K$ is quasi--compact and dominates $X$, i.e. every valuation has a center at $X$. If $X$ is non--singular, then certainly every valuation has a non--singular center at $X$. Zariski raised the following infinitesimal problem:

Basic Problem of Uniformization:

Given a finitely generated function field $K$ over $k$ and a valuation $V$ of $K$. Does there exist a projective model $X$, for which the center of $V$ is nonsingular?

From 1939 to 1944, Zariski published a sequence of papers which established among other results the following,

  • For dimension $= 1$ or $2$, Uniformization <==> Resolution.
  • If the ground field is of characteristic $0$, then Uniformization is true for any dimension.
  • If the ground field is algebraically closed and of characteristic 0, then Resolution is true for dimension 3.

    Zariski's work was obviously a summit of mathematics in the 40's and will be analyzed forever by the coming mathematicians. We would like to point out two important points in his arguments. Firstly by his reduction to hypersurfaces argument, we may consider a single equation. we may assume that $f_1 = 0$. We may call this the ``killing of $f_1$''. Note that from the time immemorial this technique has been used to solve a quadratic equation. With it Zariski showed easily that either all coefficients are isolated with respect to blow--ups, hence there is a reduction of the number of variables or there is a reduction of multiplicity $m$ which is certainly an improvement on the singularity.

    Secondly Zariski followed in his procedure the ``Perron Algorithm'' which is a plain generalization of continuous fractions. The above mentioned two techniques cannot be used in Uniformization for characteristic $p > 0$ and Resolution for higher dimensional case. The ``killing of $f_1$'' simply does not exist in characteristic $p$, or even worse, if $f_1$ accidentally is zero, Zariski's results cannot be reproduced. On the other hand, the ``Perron Algorithm'' is too rigid and too local to be used thus far in the global problem of Resolution.

    Among Zariski's outstanding students, S.S. Abhyankar and H.Hironaka had prominent contributions to this problem. In his Ph.D. thesis of 1956, S.Abhyankar used a sequence of going--up and coming--down theorems to solve the surface uniformization problem.

    Ten years later S.S.Abhyankar sharpened his algorithm for surfaces and established that for a point of a 3 dimensional hypersurface with multiplicity $<$ the characteristic of the field, the singularity can be resolved, thus adopting an Albanese map, which guarantees all multiplicities < or = (\dim)!, S.S.~Abhyankar established a resolution of 3--fold except $p = 2,3,5 .

    In 1964, H.Hironaka published a celebrated paper and solved completely Resolution for any dimensional for characteristic zero. Grothendick once claimed orally that Hironaka's work is the most complicated mathematical work. This is rightly so. After a quarter of a century, nobody has been successful in simplifying the web of inductions in Hironaka's work. This is truly amazing. Giraud once said that even consider only the 2--dimensional case. Hironaka's web of inductions is not much simplified.

    For the last few years, I was taught and influenced by H.Hironaka about the problem. I proved the `3-dimensional uniformization in characteristic p'. Then there was a striking news. In the May of 1992, M Spivakovsky announced a proof of the general case. Since then he is still writing his manuscript which has been thought to be 23 pp at the beginning. The last email received by me containing an assurance from him that all details have been straighten out. However, many people (mostly non-experts), who have access to some versions of his manuscript, judge unfavably. It is anybody's guess if the "proof" is correct.