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?
