Recall that a hypersurface $D$ in a complex manifold $X$ is called a free divisor if the Lie algebroid $\mathcal{T}_X(-\log D)$ of vector fields tangent to $D$ is locally free. This condition is equivalent to requiring that the singular locus $D_{sing}$ has codimension two in $X$ and is Cohen-Macaulay.

I am wondering what is known about an analogous situation: hypersurfaces $D \subset X$ for which $D_{sing}$ has codimension three in $X$ and is Gorenstein. Such hypersurfaces arise naturally in Poisson geometry as the degeneracy loci of generically symplectic Poisson structures. (Hence, there is a Lie algebroid present here, as well.) An example is the secant variety to an elliptic normal quintic curve $E \subset \mathbb{P}^4$, which is the degeneracy locus of one of Feigin and Odesskii's famous elliptic Poisson structures. Its singular locus is a degree-40 Gorenstein scheme supported on $E$.

I am interested in learning anything that is known about such hypersurfaces, but to single out a precise question that is relevant to Poisson geometry, I ask the following:

Which quintic hypersurfaces in $\mathbb{P}^4$ have a (possibly non-reduced) Gorenstein curve as their singular locus?

I would be happy with an answer that gives one or two examples (other than the secant variety above), and/or provides a reference to relevant literature.

  • $\begingroup$ This doesn't answer your question, but when you mention codimension 2 Cohen-Macaulay and codimension 3 Gorenstein, I immediately think of the Hilbert-Burch theorem en.wikipedia.org/wiki/Hilbert-Burch_theorem and the Buchsbaum-Eisenbud theorem jstor.org/discover/10.2307/… For the latter the global version in the EPW paper arxiv.org/abs/math/9906170 might be relevant. $\endgroup$ – Steven Sam Sep 13 '13 at 0:52
  • $\begingroup$ Thanks for the references, @StevenSam. Those papers were, indeed, part of the motivation for the question. In fact, the proof that the Poisson divisors I mentioned have codimension three Gorenstein singular loci proceeds by expressing the singular locus as the degeneracy locus of a certain skew form obtained from the Poisson structure. $\endgroup$ – Brent Pym Sep 13 '13 at 3:11

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