So this is a follow up on the post about Burch’s theorem. Namely, I’ve just learned in the last month or so that the next case of this is in Eisenbud + Buchsbaum Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. It says that the resolution of a codimension 3 Gorenstein singularity R/I with R regular has a free resolution of the form
0 —> R —> R^n —f—> R^n —> R
where f is an alternating matrix and the other arrows are given by Pfaffians of f.
Moreover, if R/J is an almost complete intersection of grade 3, then R/J is linked to a Gorenstein R/I as above and a similar type of resolution can be obtained (results of Brown, kustin, etc).
OK, this is cool, very cool.
It seems completely clear that similarly to Burch’s theorem this implies that such a singularity is unobstructed, just as in the codimension 2 Cohen-Macaulay case. To be precise, as a simple consequence of the paper we obtain:
If R = k[[x, y, z]] and R —> S is an Artinian quotient ring such that either (1) S is Gorenstein, or (2) the kernel of R —> S is generated by at most 4 elements, then the miniversal deformation space of S is a power series ring over k.
What I’d like is a reference to articles (with page and line numbers) stating exactly the above for (1) and (2). A generic reference to unobstructedness of determinantal singularities doesn’t count. I’ve googled and binged, but no luck so far. Can you help?
Or maybe this is just one of the innumerable results in our field that are so clearly true that you cannot formulate it in a paper as your paper will be immediately rejected?