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The Monge-Cayley-Salmon theorem via classical differential geometry

28 March, 2014 in expository, math.AG, math.DG | Tags: , , | by | 11 comments

Let {f: {\bf R}^3 \rightarrow {\bf R}} be an irreducible polynomial in three variables. As {{\bf R}} is not algebraically closed, the zero set {Z_{\bf R}(f) = \{ x \in{\bf R}^3: f(x)=0\}} can split into various components of dimension between {0} and {2}. For instance, if {f(x_1,x_2,x_3) = x_1^2+x_2^2}, the zero set {Z_{\bf R}(f)} is a line; more interestingly, if {f(x_1,x_2,x_3) = x_3^2 + x_2^2 - x_2^3}, then {Z_{\bf R}(f)} 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 {2}-dimensional component {Z_{{\bf R},2}(f)} is non-empty, thus defining a real surface in {{\bf R}^3}. In particular, this hypothesis implies that {f} is not just irreducible over {{\bf R}}, but is in fact absolutely irreducible (i.e. irreducible over {{\bf C}}), since otherwise one could use the complex factorisation of {f} to contain {Z_{\bf R}(f)} inside the intersection {{\bf Z}_{\bf C}(g) \cap {\bf Z}_{\bf C}(\bar{g})} of the complex zero locus of complex polynomial {g} and its complex conjugate, with {g,\bar{g}} having no common factor, forcing {Z_{\bf R}(f)} to be at most one-dimensional. (For instance, in the case {f(x_1,x_2,x_3)=x_1^2+x_2^2}, one can take {g(z_1,z_2,z_3) = z_1 + i z_2}.) Among other things, this makes {{\bf Z}_{{\bf R},2}(f)} a Zariski-dense subset of {{\bf Z}_{\bf C}(f)}, thus any polynomial identity which holds true at every point of {{\bf Z}_{{\bf R},2}(f)}, also holds true on all of {{\bf Z}_{\bf C}(f)}. 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 {Z_{{\bf R},2}(f)} is said to be ruled if, for a Zariski open dense set of points {x \in Z_{{\bf R},2}(f)}, there exists a line {l_x = \{ x+tv_x: t \in {\bf R} \}} through {x} for some non-zero {v_x \in {\bf R}^3} which is completely contained in {Z_{{\bf R},2}(f)}, thus

\displaystyle f(x+tv_x)=0

for all {t \in {\bf R}}. Also, a point {x \in {\bf Z}_{{\bf R},2}(f)} is said to be a flecnode if there exists a line {l_x = \{ x+tv_x: t \in {\bf R}\}} through {x} for some non-zero {v_x \in {\bf R}^3} which is tangent to {Z_{{\bf R},2}(f)} to third order, in the sense that

\displaystyle f(x+tv_x)=O(t^4)

as {t \rightarrow 0}, or equivalently that

\displaystyle \frac{d^j}{dt^j} f(x+tv_x)|_{t=0} = 0 \ \ \ \ \ (1)

 

for {j=0,1,2,3}. Clearly, if {Z_{{\bf R},2}(f)} is a ruled surface, then a Zariski open dense set of points on {Z_{{\bf R},2}} 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 {f: {\bf R}^3 \rightarrow {\bf R}} be an irreducible polynomial with {{\bf Z}_{{\bf R},2}} non-empty. Suppose that a Zariski dense set of points in {Z_{{\bf R},2}(f)} are flecnodes. Then {Z_{{\bf R},2}(f)} 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 {\{ (x_1,x_2,x_3): x_3 = x_1^2 - x_2^2 \}}, 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. {x_3 = x_1^2 - x_2^2 + x_2^4}) 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 {l_x} indicated by the flecnode to produce smooth curves {\gamma} on the surface {{\bf Z}_{{\bf R},2}}; 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.

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