Let be an irreducible polynomial in three variables. As
is not algebraically closed, the zero set
can split into various components of dimension between
and
. For instance, if
, the zero set
is a line; more interestingly, if
, then
is the union of a line and a surface (or the product of an acnodal cubic curve with a line). We will assume that the
-dimensional component
is non-empty, thus defining a real surface in
. In particular, this hypothesis implies that
is not just irreducible over
, but is in fact absolutely irreducible (i.e. irreducible over
), since otherwise one could use the complex factorisation of
to contain
inside the intersection
of the complex zero locus of complex polynomial
and its complex conjugate, with
having no common factor, forcing
to be at most one-dimensional. (For instance, in the case
, one can take
.) Among other things, this makes
a Zariski-dense subset of
, thus any polynomial identity which holds true at every point of
, also holds true on all of
. This allows us to easily use tools from algebraic geometry in this real setting, even though the reals are not quite algebraically closed.
The surface is said to be ruled if, for a Zariski open dense set of points
, there exists a line
through
for some non-zero
which is completely contained in
, thus
for all . Also, a point
is said to be a flecnode if there exists a line
through
for some non-zero
which is tangent to
to third order, in the sense that
for . Clearly, if
is a ruled surface, then a Zariski open dense set of points on
are a flecnode. We then have the remarkable theorem (discovered first by Monge, and then later by Cayley and Salmon) asserting the converse:
Theorem 1 (Monge-Cayley-Salmon theorem) Let
be an irreducible polynomial with
non-empty. Suppose that a Zariski dense set of points in
are flecnodes. Then
is a ruled surface.
Among other things, this theorem was used in the celebrated result of Guth and Katz that almost solved the Erdos distance problem in two dimensions, as discussed in this previous blog post. Vanishing to third order is necessary: observe that in a surface of negative curvature, such as the saddle , every point on the surface is tangent to second order to a line (the line in the direction for which the second fundamental form vanishes). This surface happens to be ruled, but a generic perturbation of this surface (e.g.
) will no longer be ruled, although it is still negative curvature near the origin.
The original proof of the Monge-Cayley-Salmon theorem is not easily accessible and not written in modern language. A modern proof of this theorem (together with substantial generalisations, for instance to higher dimensions) is given by Landsberg; the proof uses the machinery of modern algebraic geometry. The purpose of this post is to record an alternate proof of the Monge-Cayley-Salmon theorem based on classical differential geometry (in particular, the notion of torsion of a curve) and basic ODE methods (in particular, Gronwall’s inequality and the Picard existence theorem). The idea is to “integrate” the lines indicated by the flecnode to produce smooth curves
on the surface
; one then uses the vanishing (1) and some basic calculus to conclude that these curves have zero torsion and are thus planar curves. Some further manipulation using (1) (now just to second order instead of third) then shows that these curves are in fact straight lines, giving the ruling on the surface.
Update: Janos Kollar has informed me that the above theorem was essentially known to Monge in 1809; see his recent arXiv note for more details.
I thank Larry Guth and Micha Sharir for conversations leading to this post.
— 1. Proof —
Let denote the smooth points of
, then
is a smooth surface that is a Zariski open dense subset of
, and hence Zariski dense in
. We consider the projective tangent bundle
of
; this is a smooth three-dimensional manifold, which is a bundle of copies of the projective line
over
, with elements
consisting of a point
in
and the projective class of a direction
that is tangent to
at
and is non-zero. Since
and
are both irreducible varieties, it is easy to see that
is also an irreducible variety.
Inside , we consider the subset
of points
which obey the flecnode condition (1) for
. By hypothesis, the projection of
to
is Zariski dense. On the other hand,
is clearly an algebraic set. Thus the dimension of
is at least
, and there is at least one component whose projection to
is two-dimensional (i.e. is dominant). In particular we can find an irreducible algebraic surface
in
whose projection to
is open dense (not just in the Zariski sense, but also in the differential geometry sense). By removing the singular points of
, we may assume that
is a smooth surface.
We now claim that the projection map is generically a local diffeomorphism, thus
has full rank for a Zariski dense set of points
in
. This is a simple consequence of Sard’s theorem, but for our purposes it is also instructive to see an ODE proof: if
fails to have full rank generically, then it must have rank one generically or rank zero generically. If it has rank one generically, one can use the Picard existence theorem to locally foliate an open dense subset of
by curves
with the property that for each
, the derivative
lies in the kernel of
, so that if we write
, then
for all
, and so
is constant; thus the curves each lie in a single fibre of
. This locally describes
as a one-dimensional smooth family of curves inside the fibre of
, and so the image
is locally one-dimensional, contradicting the two-dimensional nature of
. A similar argument works when
has rank zero generically.
Since is a local diffeomorphism generically, we may apply the inverse function theorem to conclude that on an open dense subset of
, we can locally invert this map, which in particular gives smooth local maps
from open subsets of
to unit tangent vectors
at
such that the flecnode condition (1) is satisfied for all such
and
.
By the Picard existence theorem, we may thus locally foliate by curves
with the property that
for all ; thus
has unit speed and is always tangent to a flecnode direction. Thus, by (1) we have
for . Expanding this out in coordinates by the chain rule (and using the usual summation conventions), using
to denote the components of
, and
to denote the first partial derivatives of
for
,
to denote the second partial derivatives, and so forth, we have
We can obtain further differential equations by differentiating the above equations in . For instance, if we differentiate (3) in
we obtain
and hence by (4)
Similarly, if we differentiate (4) in we obtain
and hence by (5)
Finally, if we differentiate (6) in we obtain
and hence by (7)
The equations (3), (6), (8) have a simple geometric interpretation: the first three derivatives are all orthogonal to the gradient
. Generically, this gradient is non-zero, and we are in three dimensions, so we conclude that
are always coplanar. Equivalently, the torsion of the curve
vanishes, and hence the curve
is necessarily planar (locally, at least). Another way to see this is to start with the identity
where is the cross product, and conclude that
is a scalar multiple of
whenever it is non-vanishing, which by Gronwall’s inequality shows that
has fixed orientation whenever it is non-vanishing.
So there is a plane in
in which
locally lies. If
vanished on this plane, then
, being irreducible, would be just
and we would be done, so we may assume that
is non-vanishing here, thus
is at most one-dimensional. On the other hand, (3), (6) show that
are both orthogonal to the gradient of
restricted to
, which is generically non-zero; as we now only have two dimensions, this implies that
are parallel. Thus the curvature of
now also vanishes, which implies that
is a straight line. Hence we have locally foliated at least a small open neighbourhood in
by straight lines, which ensures that
is ruled as desired.
11 comments
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28 March, 2014 at 1:12 pm
Nets Katz
Compare with Art 437 of Salmon’s Treatise of Analytic Geometry in Three Dimensions volume 2 and is on pages 18-20 of that volume.
28 March, 2014 at 1:44 pm
Nets Katz
I mean his only crime is that he uses p,q, r,s,t, \alpha \beta, \gamma, \delta to denote components of the 3-jet of the surface, and now 160 years later, no one will use his textbook.
28 March, 2014 at 1:54 pm
belwas
dear terry please change: ‘I think Larry Guth and Micha Sharir’ to I thank
[Corrected, thanks – T.]
28 March, 2014 at 8:26 pm
tomcircle
Reblogged this on Math Online Tom Circle.
29 March, 2014 at 9:33 pm
allenknutson
I don’t understand the statement near the top, “We’ll assume the 2-d oomponent is nonempty, and therefore f is absolutely irreducible.” What about f = x_1 x_2 x_3? Do you maybe want irreducible => absolutely irreducible?
[f is assumed irreducible in the first line of the post – T.]
30 March, 2014 at 4:46 am
tomsim
Secondlast Intro Paragraph contains a “clsasical” typo.
2 April, 2014 at 11:51 am
Mustafa
Is there a non-commutative analog of the Cayley-Salmon theorem?
3 April, 2014 at 11:59 am
arch1
In paragraph 2 do you mean “…Zariski open dense *set* of points…”?
[Corrected, thanks – T.]
25 September, 2015 at 8:08 pm
Anonymous
Dear Tao, surely I don’t understand, but the example after theorem 1, the saddle, seems doubly ruled, like z=xy.
25 September, 2015 at 9:02 pm
Terence Tao
Oh, that’s a bit unfortunate. But one can perturb the saddle a little bit while still keeping it negatively curved (say, near the origin) while destroying its ruledness; I’ve added a remark to this effect in the post.
30 December, 2020 at 1:06 pm
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