# Linearization instability and singular points of algebraic varieties

In a well known 1973 paper, Fischer and Marsden pointed out (with similar, contemporary remarks made in the physics literature by Brill and Deser) that the space of solutions of some non-linear partial differential equations, like the Einstein or Yang-Mills equations, on a manifold with compact Cauchy slices cannot be manifold (of an appropriate infinite dimensional kind). This phenomenon was given then name linearization instability; I'll explain it in more detail below.

As far as I can tell, this terminology is only used in the literature that followed this observation. However, the actual mathematical phenomenon is very similar to the fact that algebraic varieties are also not manifolds, since they possess singular points. My question is this: what terminology is used in algebraic geometry for the various aspects of this phenomenon? I'll be more specific below.

Given a non-linear PDE, $E[u] = 0$, where $u$ is the unknown function, defined on some manifold $M$. Let its linearization about a background solution $E[U]=0$ be $E_U[u] = 0$, that is, $$E[U+\varepsilon u]=\varepsilon E_U[u] + O(\varepsilon^2).$$ Let $X$ denote the set of solutions of the full equation (exact solutions) and let $\ker E_U$ denote the set of all solutions of the linearized equation (linearized solutions). The background solution $U$ is said to be linearization stable if each linearized solution $u\in \ker E_U$ can be extended to a 1-parameter family $v_\varepsilon = U + \varepsilon u + O(\varepsilon^2)$ of exact solutions, $E[v_\varepsilon] = 0$. If $U$ is linearization stable, then one can show that the space $X$ of exact solutions that are close to $U$ is an (infinite dimensional) manifold, modeled on the space $\ker E_U$.

On the other hand, it is possible that there exists a function $Q[u]$ (typically involving some integral of a local expression over a submanifold of $M$) such that a linearized solution $u\in \ker E_U$ cannot be extended to a 1-parameter family $v_\varepsilon = U + \varepsilon u + O(\varepsilon^2)$ of exact solutions unless $Q(u)=0$. If such a non-trivial function $Q[u]$ exists, the background $U$ is said to be linearization unstable. In the examples of Einstein or Yang-Mills equations, $U$ happens to be linearization unstable if it possess some symmetries and the corresponding $Q(u)$ is quadratic. One can then show that the space $X$ of exact solutions of $E[v]=0$ around $v=U$ is not a manifold, since it has neighborhoods that are not homeomorphic to any vector space (in particular not $\ker E_U$) but rather to a conical subset of $\ker E_U$ defined by $Q[u]=0$.

Sometimes it is possible to identify a set of functions $S_i[u]$ such that the intersection of the zero set $S_i[u]=0$, for all $i$, with the space $X$ of exact solutions consists only of linearization unstable backgrounds. In the example of Einstein's equations, the $S_i[u]=0$ could consist of the Killing equations, which identify metrics with non-trivial symmetries.

As I've already mentioned, a linearization unstable point in $X$ is very similar to the singular point of an algebraic variety. Namely, consider the following analogy. Instead of a function on a manifold, $u$ is a finite dimensional vector. Instead of a PDE, $E[u]=0$ is a polynomial equation, which defines an algebraic variety $X$. A linearization unstable point $U\in X$ is, I believe, called a singular point of $X$. For instance, we could have $u=(x,y,z)$, $E[u]=x^2+y^2-z^2$, $U=(0,0,0)$, $E_U[u] = 0$, $Q[u] = x^2+y^2-z^2$, $S[u] = z$.

Could someone provide a translation dictionary into the appropriate terminology used in algebraic geometry (preserving grammatical structure if possible)?

• linearization sability/instability --- ?
• linearization stable/unstable point $U\in X$ --- ?
• linearized solutions $\ker E_U$ --- ?
• functions like $Q[u]$ --- ?
• the zero set $Q[u]=0$ in $\ker E_U$ --- ?
• functions like $S_i[u]$ --- ?
• the zero set $S_i[u]=0$ --- ?

Also, do functions like $S_i[u]$ play a significant role in blowing up singular points?

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Maybe the best answer I can give is ''learn about deformation theory''. But I will take a shot at writing the dictionary you request, in the case where $E(u)$ is a polynomial function

• linearization stability <--> smoothness / instability <--> singularity
• linearization stable/unstable point <--> singular / smooth point
• linearized solutions <--> Zariski tangent space
• functions like $Q[u]$ <--> obstructions
• the zero set $Q[u]=0$ <--> (Edited.) the tangent cone
• functions like $S_i[u]$ <--> the functions $\partial E/\partial u$, or maybe it is better to say, a certain Fitting ideal of the sheaf of Kahler differentials on $X$
• the zero set $S_i[u] = 0$ <--> on $X$, you would I guess call it the singular locus; in the whole ambient space I don't know.

Regarding ''learn about deformation theory'', the point is that you started with some very explicit algebraic variety $X$ and then decided to think about its structure near a point. But deformation theory lets you think about the following thing: say you want to think about the space of all algebraic ''solutions'' of some sort near a given ''solution'', but to a less explicit problem like ''describe all holomorphic maps from a genus 5 curve into projective space'', which of course can also be written as a PDE. It may be difficult or worse to construct such a space globally, but still you can start to think about the local infinitesimal structure near a given solution using infinitesimal methods; and then there are very powerful algebraic tools (the Artin approximation theorem) which let you under good conditions construct an algebraic neighborhood of your given solution.

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Thanks, this is very helpful! What is a good (perhaps on the elementary side) reference for deformation theory? –  Igor Khavkine Feb 24 '13 at 10:29
unfortunately I'm not sure if that exists. You could try Hartshorne's notes math.berkeley.edu/~robin/math274root.pdf –  Vivek Shende Feb 24 '13 at 14:44
It took me a while, in between things, to learn about all this algebraic geometry terminology. But everything you wrote is pretty much spot on! I would only add that the zero set $Q[u]=0$ is just the tangent cone of $X$ at a given point. –  Igor Khavkine Aug 23 at 7:02