Higher dimensional nodes

Question

A node on a curve is a singular point that locally looks like the intersection of two lines. I think the precise way to say this is that $p \in X$ is a (closed?) point on a scheme $X$ (of finite type over a field $k$?), then the completion of the local ring at $p$, $\widehat{\mathcal{O}}_{X,p}$ should be isomorphic to $k[[x,y]]/(xy)$ (the completion for the intersection of two lines).

The first question is whether this is correct.

The second question is whether you get the n-dimensional version of a node by requiring the completion of the local ring to look like intersection of $n+1$ coordinate planes: $k[[x_0,x_1,...,x_n]]/(x_0 x_1\cdots x_n)$.

The third question is: what exactly does it mean for such a singularity to be isolated? This is easy to imagine over $\mathbb{C}$ (there is analytic neighborhood containing no other singularity), but how to say this in something like the Zariski topology?

Answer 1

[Edit: Over an algebraically closed field] a node should be an isolated hypersurface singularity whose (projectivized) tangent cone is a nondegenerate quadric. This means that in local coordinates the equation has no linear part, and the quadratic part is nondegenerate.

For a curve your definition is equivalent to this one so it is correct. In higer dimensions it is very different. In fact your hypersurface has a non-isolated singularity (for instance, all points in the coordinate lines are singular).

For the last question, the singular points of an algebraic set (or variety, or scheme) are a Zariski closed subset, which splits in irreducible components. A singular point is isolated if it is one of the irreducible components (and it is not embedded, ie, it does not belong to any other irreducible component, but this has sense only in the scheme-theoretic case).

For the non algebraically closed field case, see BCnrd's comment and reference!

Answer 2

With regards to your second question, I think it depends on what properties you want your higher dimensional node to keep. This I suppose answers the (slightly different) question of "what is a higher dimensional analog of a node?" (and for simplicity, as already pointed out, I'll work over an algebraically closed field).

The singularity you describe is an example of simple normal crossings hypersurface. In many cases, it is a perfectly good generalization of a node. In other cases, you might want to allow more singularities or (not allow your simple normal crossings hypersurface at all). Your higher dimensional analogs of nodes might be hypersurfaces, or have isolated singularities, depending on context.

For example, one place where nodal curves show up is when doing generic projections of smooth curves in $\mathbb{P}^n$ to $\mathbb{P}^2$ (see Hartshorne, Chapter IV, Section 3). In general, you can take a $d$ dimensional projective variety and generically project it to $\mathbb{P}^{d+1}$. Such generic projections will be something called ''seminormal'' (see a paper by Greco and Traverso), and they will also be Gorenstein (they are hypersurfaces). These conditions are equivalent to being a node in dimension 1. On the other hand, these conditions on a singularity are not equivalent to the singularity being a generic projection in general. Rob Lazarsfeld also discusses generic projection singularities a little bit in his ''Positivity'' book.

Another place that nodal curves show up is in the usual compactifications of moduli spaces of curves. If you look at higher dimensional varieties, then presumably the correct generalization of a node is then something called ``semi-log canonical singularities''. This is again a distinct notion from these semi-log canonical hypersurfaces, see for example Rob Lazarfeld's book and also the dissertation of Davis Doherty (University of Washington, 2006).