Zariski surgically dissecting a flawed proof

The author presents in this paper an investigation which, he claims, constitutes a complete proof of the famous conjecture that every algebraic variety can be transformed birationally into a variety which is free from singularities. He restricts himself to the case of the complex domain. [In this case, and more generally, in the case of ground fields of characteristic zero, the conjecture has been proved so far only for varieties of dimension $\leq 3$; for the case of dimension 3 see the reviewer’s paper “Reduction of the singularities of algebraic three-dimensional varieties,” [Ann. of Math. (2) 45, 472–542 (1944); MR0011006 (6,102f); this paper will be referred to in this review as RS3].] The author has published a number of other papers on the same subject, in particular a long memoir entitled “Le problème général de la réduction des singularités d’une variété algébrique” [Mém. Soc. Roy. Sci. Liège (4) 9, no. 2 (1949); MR0035487 (11,740c)]. The present paper is described in the introduction as a decisive simplification of the proof given in the above cited memoir. It is stated that the simplification is due chiefly to the author’s recent discovery of what he calls “elementary transformations.” [These transformations, under the name of “monoidal transformations,” have been introduced and fully studied by the reviewer on pages 532–542 of his paper (referred to in the sequel as FBC) “Foundations of a general theory of birational correspondences,” [Trans. Amer. Math. Soc. 53, 490–542 (1943); MR0008468 (5,11b)]; they have been extensively used by the reviewer in RS3.] The monoidal (or “elementary”) transformations are indeed elementary, in more than one sense, and while the reviewer has found them a useful tool for the resolution of the singularities, he finds it rather strange that they should have had such a tremendous effect on the author’s “proof,” for the deeper difficulties of the resolution problem can be affected by such straightforward tools only to a very moderate extent. At any rate, it is stated that his discovery of monoidal transformations has enabled the author to condense “en quelques heures” the 139 pages of his cited memoir to five pages of a note entitled “Méthode simplifiée de réduction des singularités d’une variété algébrique” [Acad. Roy. Belgique. Bull. Cl. Sci. (5) 35, 880–885 (1949); MR0035488 (11,740d)]. According to the author, this short note contained the complete solution of the problem, so much so that he felt that “pour moi, le problèmé était bel et bien résolu, mais il me restait à faire admettre ma solution.” The present more elaborate version of that note has therefore been written chiefly for the purpose of convincing the unbelievers, and the author acknowledges his indebtedness to van der Waerden who, in a series of conferences with the author (“… une dizaine de discussions de deux heures chacune, suivies pour chacun de nous de deux ou trois jours de réflection…”), has closely and critically scrutinized every single detail of the present “proof” (“… et il prit dès lors la peine de passer au crible de son esprit critique bien connu les douze pages du texte que je lui apportais, littéralement ligne par ligne”).

Before describing and discussing the author’s reasoning, a few general remarks about the paper will be in order. Its language is “geometric.” The geometric language, when it is not based on a carefully prepared algebraic basis, is never explicit or convincing in algebraic geometry. On this ground alone the author’s “proof” could be dismissed as incomplete, for in scientific work it is right to hold every author guilty until he proves himself innocent. However, out of consideration for the importance of the problem, and because of the author’s implied belief that his work has not been duly evaluated, this reviewer has reversed his attitude and has read the paper on the assumption that it is the reader who is to be held guilty until he proves himself innocent. In practice that meant that in cases of doubtful points, the reviewer has made an effort to either (a) complete the proof himself or (b) find precisely what is wrong with the proof or (and) (c) find a counterexample. With this approach, the geometric language of the author has the effect of shifting a good deal of the burden from the author to the reader, for in many cases it was at least as difficult to accomplish (a) or (b) or (c) as it was for the author to find his incomplete proofs. However, the problem of resolution of singularities presents not only difficulties of rigor but also difficulties of perception which lie much deeper. It is for this reason that the reviewer was willing to give the author the benefit of the doubt on the formal side of the treatment. He was mainly interested in finding out how the author has dealt with the irreducibly difficult core of the problem. He was forced to conclude that lack of perception of the serious (and not obvious) complications that can (and do) arise in the course of the reduction process have prevented the author from coming to grips with the real difficulties of the problem. And so it happens that the two main pillars of his “proof”—(a) the theorem on first polars on p. 314 (“qui est d’une importance capitale pour la suite”) and (b) the “raisonnement fondamental” on p. 317—are represented by statements of which the first is false and the second is far from having been proved (see theorems ${\rm A}’$ and ${\rm B}’$ below; as a matter of fact also ${\rm B}’$, in the general form stated at the bottom of p. 318, is false). As a consequence, the following can be said about the two other statements (see theorems A and B below) which are made by the author and whose logical sum implies the resolution theorem: A has not been proved because its “proof” is based on ${\rm A}’$; B has not been proved because its “proof” is based on A and ${\rm B}’$. On the whole, the author’s contribution merely scratches the surface of things: it belongs to the prenatal phase of the problem, a phase familiar to anybody who has given this problem any thought. (We pass without comment the author’s astonishing remark that his “methods” can be extended “without difficulties” to perfect fields of characteristic $p\neq 0$, it being quite superficially clear that they could not possibly be so extended.)

Let $V$ be a $k$-dimensional irreducible variety in a projective space $S_q$. To resolve the singularities of $V$ one will begin by applying to $S_q$ a monoidal transformation $T$ whose center is a suitable irreducible nonsingular variety $W$ belonging to the singular variety of $V$. The effect of $T$ is as follows: (a) it transforms birationally $S_q$ into a nonsingular variety $B_q{}’$, immersed in some projective space $S'{}_{n’}$; (b) it transforms birationally $V$ into a variety$V’$ immersed in $B_q{}’$; (c) it blows up $W$ into an irreducible $(q-1)$-dimensional subvariety $\Gamma’$ of $B_q{}’$; this variety $\Gamma’$ is free from singularities and carries a ruling of linear spaces $\Gamma_P{}’$ of dimension $q-1-\rho$ (where $\rho=\dim W$) which correspond to the individual points $P$ of$W$ (see FBC and RS3). In a second step of the resolution process, $V’$ and $B_q{}’$ will replace the original $V$ and $S_q$, and one will apply to $B_q{}’$ a monoidal transformation $T’$ whose center is a suitable irreducible nonsingular variety $W’$ belonging to the singular variety of $V’$ (the steps just described are identical to those used in RS3). We shall therefore assume that we are given, to begin with, a nonsingular $q$-dimensional variety $B\ (q\geq k+1)$ containing $V$and contained in some projective space $S_n\ (n\geq q)$. If $k=n-1$ (whence $B=S_n$) it is well known what is meant by a first polar $F$ of $V$. If $k<n-1$, a first polar of $V$ is defined as a first polar of any hypercone which projects $V$ from an $S_{n-k-2}$. The set $[F]$ of first polars of $V$ is an irreducible algebraic system (linear, if $k=n-1$); its base locus is the singular variety of $V$. The trace of $[F]$ in $B$, i.e., the set of intersections $F\cap B$, is an irreducible algebraic system $[\Phi]$ of $(q-1)$-dimensional subvarieties of $B$, and also the base locus of $[\Phi]$ is the singular variety of $V$. If $V$ is a monoidal transformation of$B$, of the type described above, one defines in an obvious fashion the $T$-transform $[\Phi’]$ of the system $[\Phi]$; this will be then an irreducible algebraic system of $(q-1)$-dimensional subvarieties of $B’\ (=B_q{}’)$ (since $W$ is a base variety of $[\Phi]$, particular members of $[\Phi’]$ may contain $\Gamma’$ as a component, to a suitable multiplicity). This process is to be repeated as long as the successive transforms $V’,V”,\cdots$ of $V$ continue to have singularities. One is thus led to a sequence $\{V^{(i)},B^{(i)},[\Phi^{(i)}]\}$. At each stage, the next, $(i+1)$th member of the sequence depends on the choice of the subvariety$W^{(i)}$ of the singular variety of $V^{(i)}$. For the purposes of this review it must be clearly understood that $[\Phi’]$ no longer has the same relationship to $V’$as $[\Phi]$ has to $V$, i.e., $[\Phi’]$ is not the $B’$-trace of the system of first polars of $V’$ (in relation to the ambient projective space $S'{}_{n’}$ of $B’$). The author then proposes to prove the following two theorems: A. Every singular point of $V^{(i)}$ is a base point of $[\Phi^{(i)}]$. B. By a suitable choice of the centers$W^{(i)}$ of the successive monoidal transformations $T^{(i)}$ it is possible to obtain, after a finite number of steps, a transform $V^{(i)}$ of $V$ such that the system $[\Phi^{(i)}]$ has no base points.

It is clear that A and B together would suffice to establish the theorem of resolution of singularities. [The idea of using the first polars and studying their behavior under quadratic transformations (in the case of algebraic surfaces) goes back to B. Levi, and was used in one form or another in every investigation dealing with theresolution problem. It may be noted that theorem B has been proved by the reviewer in the case $q=3$, for arbitrary systems $[\Phi]$ of surfaces; see RS3, Theorem$7’$, p. 531.] The essential ingredient of the author’s “proof” of A is the following theorem: ${\rm A}’$. The generic first polar $F$ of $V$ has an $(s-1)$-fold point at every (proper or improper) $s$-fold point of $V$ (“improper” means “infinitely near”). Or, in less mysterious language: if a (proper) point $O^{(i)}$ of $V^{(i)}$ is$s$-fold for $V^{(i)}$, then it is $(s-1)$-fold for the generic $\Phi^{(i)}$. The “proof” of ${\rm A}’$ is given only under the following special conditions: $k=n-1$and the monoidal transformations used in the reduction process are all locally quadratic, i.e., their centers are points. The general case is dismissed in two lines by the mere statement that the proof is similar. Now while we shall see in a moment that ${\rm A}’$ is false even under these special conditions, it may be explained now that the general case presents additional difficulties which the author does not perceive. If the center $W$ of $T$ is of positive dimension then it may well happen that$V’$ will contain a generator $\Gamma_P{}’$ of $\Gamma’$ which corresponds to some special point $P$ of $W$. Whatever conclusions one arrives at in the special case of locally quadratic transformations, these conclusions cannot be automatically used, in the general case, for making statements concerning the behavior of $V’$at $\Gamma_P{}’$; they only give information about the behavior of $V’$ at subvarieties $W’$ which correspond to the whole of $W$ (and not to proper subvarieties of $W$). [For a method of dealing with this phenomenon, see RS3.] Furthermore, if $k<n-1$ then the polars $\Phi$ are defined in terms of the general projections of$V$ (in its ambient space $S_n$), and in that case it is not at all clear how the transforms $\Phi’$ of $\Phi$ are related to the polars of the general projections of $V’$(in its ambient space $S'{}_{n’}$) and why this matter should have been dismissed in a few words.

We now come to “theorem” ${\rm A}’$ and accept the special conditions under which the author “proves” it. It is not difficult to give counterexamples, even in the simplest case of plane algebraic curves $(k=1,q=n=2)$, for “theorem” ${\rm A}’$ is “almost always” false. Here is an example. Let $V$ be the plane curve $f(x,y)=0$, where $f(x,y)=y^3-x^5$. Then $V$ has a triple point at the origin $O$ and it has a double point at the improper point $O’$ infinitely near $O$ in the direction $y=0$. By the locally quadratic transformation $T\colon x=x’$, $y=x’y’$, the curve $V$ is transformed into the curve $V’$ given by the equation $x^{\prime 2}-y^{\prime 3}=0$, the improper point $O’$ now being represented by the origin $x’=y’=0$. The generic polar $\Phi$ of $V$ is $uf_x{}’+vf_y{}’+w(xf_x{}’+yf_y{}’-5f)=0$, i.e.,$3vy^2-5ux^4-2wy^3=0$. The $T$-transform of $\Phi$ is $3vy^{\prime 2}-5ux^{\prime 2}-2wx’y^{\prime 3}=0$, and this has at $O’$ a singular point of multiplicity 2 (and not 1, as is claimed in ${\rm A}’$). The error in the author’s proof is the following (we use his notations, pp. 313–314): it is taken for granted that if $P$ is generic with respect to $G$, then $G’$ will be such that also $P’$ will be generic with respect to $G’$. The fact is that in most cases $P’$ will not be generic with respect to $G’$. The author commits here the same error that has been committed once before in the “proof” of the following incorrect statement: “the composition of the singularity of a generic projection of an algebraic curve in $S_3$ is the same as the composition of the corresponding singularity of the space curve itself.” The error in this reasoning has been pointed out long ago by the reviewer (see p. 12 of the reviewer’s book “Algebraic Surfaces” [Springer, Berlin, 1935]). From the moment that $P’$ is not necessarily generic with respect to $G’$ and hence $H_1$ is not necessarily a generic first polar of $G’$, the multiplicity of $H_1$ at$O_1{}’$ may be greater than $s_1-1$, the lemma on pencils of hypersurfaces given on p. 313 is then not applicable, and the entire proof on p. 314 breaks down. What is even more serious is that as soon as the generic $\Phi’$ has at $O_1{}’$ a point of multiplicity $\geq s_1$, then in a second quadratic transformation the fundamental surface $\Pi”$ will detach itself at least $s_1$ times from the transform of $\Phi’$ while it will detach itself precisely $s_1-1$ times from the transform of the generic polar of $G’$. Thus already after two steps we find that further investigation would be necessary before one can make any assertion concerning the multiplicities of $\Phi”$ at the singular points of $G”$, as compared with the multiplicities which the first polars of $G”$ have at these points. What happens after $i$steps is altogether nebulous.

We now pass to theorem B. Part of the “proof” of this theorem, for dimension $k$, presupposes theorem A for all dimensions $\leq k-1$, and hence already on this ground the proof is incomplete. That part of the “proof” which is independent of theorem A is based on a theorem which the author states under the heading: “raisonnement fondamental” (p. 317). We shall explain the gap in the author’s proof, and for simplicity we shall consider the case $k=3$, $q=n=4$. We shall also assume for simplicity that the singular variety of $V$ (and hence also the base locus of $[\Phi]$) is of dimension 1. Let $[\Phi_1]$ be the characteristic system of $[\Phi]$, i.e., the systems of surfaces which are intersections of pairs of $\Phi$‘s. We shall also assume that the characteristic system $[\Phi_2]$ of $[\Phi_1]$ (from which the base curves of $[\Phi_1]$ have been deleted) has no base points (in the notation of the author, we have assumed $\sigma=0$, $k_\sigma=1$). Let $\gamma$ be an irreducible component of the base curve of $[\Phi_1]$. We assume that $\gamma$ is free from singularities. We take now the curve $\gamma$ as the center of our monoidal transformation $T$ and we denote by $[\Phi_1{}’]$ the $T$-transform of the system of surfaces $[\Phi_1]$. The “raisonnement fondamental,” in this special case, is the following assertion: ${\rm B}’$. Let $\gamma’$ be an irreducible curve in $\Gamma’$ which belongs to the base curve of the system $[\Phi_1{}’]$. If the curve $\gamma$ is $s$-fold for the generic $\Phi$ and $t$-fold for the generic $\Phi_1$ and if $m$ is the intersection multiplicity of $\Phi$ and $\Phi_1$ at $\gamma$, then the intersection multiplicity at $\gamma’$ of a generic $\Phi’$ and a generic $\Phi_1{}’$ is $\leq m-st$ (whence, at any rate, $<m$).

In the proof the author takes it for granted that $\gamma’$ meets each generator $\Gamma_P{}’$ of $\Gamma’\ (P\in\gamma)$. In other words, he assumes that $\gamma’$ corresponds to the whole of $\gamma$. He does not consider the possibility that $\gamma’$ may correspond to some special point $P$ of $\gamma$ (in which case $\gamma’\subset\Gamma_P{}’$ and $\gamma’$ does not meet $\Gamma_Q{}’$ if $Q\neq P$). To such a curve $\gamma’$ the author’s reasoning is not applicable. The reviewer has asked himself the following question: May it be that the presence of a base curve $\gamma’$ contained in $\Gamma_P{}’$ implies that $P$ is a base point of $[\Phi_2]$ (contrary to the assumption that $[\Phi_2]$ has no base points)? If that is so, it must be proved. The reviewer doubts that that is so. He knows that the above possibility can actually arise in the case of a linear system of hypersurfaces $[\Phi]$ in $S_4$ whose successive characteristic systems $[\Phi_1]$ and $[\Phi_2]$ satisfy all the conditions (as to their base varieties) stated above; and he does not see why the fact that we are dealing with polar hypersurfaces should make any difference. At any rate, the author correctly remarks that his reasoning makes no use of any special properties of the polars $\Phi$ and that even the fact that the $\Phi$‘s are complete intersections plays no role in the proof. He therefore reformulates his “raisonnement fondamental” as a general theorem concerning arbitrary pairs of varieties $\Phi$ and $\Phi_1$ in B, of dimensions $q-1$ and $q’$ respectively, $q’\leq q-1$ (see end of p. 318). Now this generalization is definitely false, even if $\Phi$ and $\Phi_1$ are complete intersections. Consider, for instance, the following case: $\Phi$ and $\Phi_1$ are respectively a hypersurface and a surface in $S_4$, having a common point $P$ which is an ordinary $n$-fold point of $\Phi$ and an ordinary $\nu$-fold point of $\Phi_1\ (n>1,\nu>1)$; we also assume that $\Phi$ and $\Phi_1$ have in common a line $\gamma$ through $P$ and that the intersection multiplicity at $\gamma$ is 1 (whence $m=s=t=1$). If $\gamma$ is taken as the center of our monoidal transformation $T$, then the following facts can be established in a straightforward manner: (a) the plane $\Gamma_P{}’$ belongs to the three-dimensional variety $\Phi’$, the latter passing through $\Gamma_P{}’$ with multiplicity $\geq n-1$; (b) $\Gamma_P{}’$ meets $\Phi_1{}’$ in a curve $\gamma’$ which is at least $(\nu-1)$-fold for $\Phi_1{}’$. It follows that the intersection multiplicity of $\Phi’$ and$\Phi_1{}’$ at $\gamma’$ is $\geq(n-1)(\nu-1)\geq 1>0=m-st$. This example shows in fact that the intersection multiplicity of $\Phi’$ and $\Phi_1{}’$ at a curve $\gamma’$ belonging to a generator $\Gamma_P{}’$ of $\Gamma’$ may have no particular relationship to either $m-st$ or $m$.