I am looking for a paper of Weil that is published under a pseudonym, in which he proves a statement along the lines of: a singular algebraic variety cannot be deformed into a nonsingular one.


Thanks in advance.



Weil's paper under a pseudonym on deforming singular varieties
this pseudonomous letter mentioned by Jim Humphreys is too amusing not to summarize here:


R. Lipschitz (Ann. of Math. 69, 1959, 247-251)


*reprinted in A. Weil, Collected Papers, volume II (Springer, 1979):*


![](https://ilorentz.org/beenakker/MO/Lipschits.png)




---


and for the record, here is a translation of the 1957 anonymous note in Italian:



> 
> A correspondent, who wishes to remain anonymous, writes as follows:
> 
> 
> --- A very well known conjecture by F. Severi (Rend. Pal. 28 (1909), p. 45) asserts that "every variety with multiplicities in some points
>  can be considered as the limit of one without singularities, belonging
>  to the same space."
> 
> 
> According to a note distributed by Nancago from "United Press" agency,
>  this hypothesis from that famous author would have been refuted by the
>  respectable French geometer René Thom, based on the example of cone of
>  3d order in the space $S\_6$ and following quite delicate topological
>  considerations.
> 
> 
> It might not displease the readers of your esteemed journal to find
>  here an elementary geometric treatment of Thom's example. In fact, we
>  will determine all the varieties of 3d order in any space $S\_n$. From
>  this we will obtain, as an immediate consequence, the falseness of the
>  hypothesis in question.
> 
> 
> Let $V\_r$ be a variety of 3d order in the space $S\_n$, of dimension $r<n-1$, 
>  not contained in a hyperplane. Let $C$ be the cone that projects $V\_r$
>  from an arbitrary simple point $M$ of $V\_r$; let $M'$ be another point
>  in $V\_r$, simple in the cone $C$. The cone $C$ is of second order;
>  hence its projection from $M'$ is a linear variety of dimension $r+1$,
>  so that $V\_r$ is contained in a linear variety of dimension $r+2$. It
>  follows that $r=n-2$ and $C$ has dimension $n-1$. The same holds for
>  cone $C'$, that projects $V\_r$ from $M'$. The cones $C$ and $C'$ are
>  distinct, hence $M'$ is simple in cone $C$; their intersection is
>  therefore a reducible variety of 4th order, broken in $V\_r$ and a
>  linear variety. Let $A=0,B=0$ be the equation of this variety. Then
>  the equations for the cones $C$, $C'$ can be written in the form
>  $$(1)\qquad\qquad AP=BQ,\;\;AP'=BQ',$$ where we denote by $P,Q,P',Q'$ four linear
>  forms homogeneous in the space coordinates. Let $s$ be the number of
>  independent forms among the six forms $A,B,P,Q,P',Q'$; this number is
>  at least 4, because, if it where 2, the cones $C,C'$ would not be
>  reducible; if it where 3, then their intersection would be broken in
>  four distinct or non-distinct varieties. The possible values for $s$
>  are therefore 4,5 and 6.
> 
> 
> Let $L$ be the linear variety of dimension $n-2$, defined by the
>  equations $A=B=0$, $P=Q=P'=Q'=0$. Each linear variety that projects
>  from $L$ a point in the intersection of $C$ and $C'$ is contained in
>  it, as one can see immediately from equations (1). Projecting $V\_r$
>  from $L$ one therefore obtains a variety $W$ of 3d order
>  and dimension $s-3$ in a space $S\_{s-1}$; and $V\_r$ is nothing else
>  than the cone projecting $W$ from $L$. In the space $S\_{s-1}$ one can
>  take as homogeneous coordinates the $s$ independent forms out of the
>  forms $A,B,P,Q,P',Q'$; therefore the equations (1) define a variety of
>  4th order, broken in $W$ and a linear variety. Now we can distinguish
>  three cases:
> 
> 
> (a) $s=6$: $W$ is the variety $W\_3$ of Segre embedded in $S\_5$, the
>  bi-rational image without exceptions of the product of a plane and a
>  line; as it is known, this has no points with multiplicities.
> 
> 
> (b) $s=5$: $W$ is the hyperplane section of $W\_3$. It is easy to see
>  that all the irreducible hyperplane sections of $W\_3$ are projectively
>  equivalent among each other; denoting by $W\_2$ one of these, it is
>  also a rational variety without points having multiplicities.
> 
> 
> (c) $s=4$: $W$ is the well known cubic rational $W\_1$ embedded in
>  $S\_3$.
> 
> 
> In this way we have demonstrated that every variety of 3d order
>  belongs to one of the following types:
> 
> 
> 1. a variety of dimension $r$, contained in a linear variety of dimension $r+1$;
> 2. the three rational varieties $W\_3$, $W\_2$, $W\_1$ enumerated above;
> 3. the projective cone of one of these three varieties from a linear variety of arbitrary dimension.
> 
> 
> Evidently, a variety of type two or three can not be the limit of a
>  variety of type one, and hence a variety of type three, of dimension
>  greater than 3, can not be the limit of a variety without
>  multiplicities in some points.
> 
> 
> 


