# Calculating a dual variety to a Chow variety

I am unfamiliar with algebraic geometry, yet I am faced with calculating three special cases of the following (the full text can be found at http://arxiv.org/abs/1107.4659)

The $n^{\times d}$-hyperdeterminant of a symmetric tensor of degree $d \ge 2$ on $n \ge 2$ variables splits as the product $\prod_\lambda \Xi_{\lambda,n}^{M_\lambda}$ where $\Xi_{\lambda,n}$ is the equation of the dual variety of the Chow variety $Chow_\lambda \mathbb{P}^{n-1}$ when it is a hypersurface in $\mathbb{P}^{\binom{n-1+d}{d}-1}$, $\lambda = (\lambda_1, \ldots , \lambda_s)$ is a partition of $d$, and the multiplicity $M_\lambda = \binom{d}{\lambda_1, \ldots , \lambda_s}$ is the multinomial coefficient.

This is theorem 1.1 of the paper by Oeding, L., (2012) titled “Hyperdeterminants of Polynomials” in Advances in Mathematics, 231, 1308-1326. The theorem is followed by this explanation:

“Geometrically, this theorem is essentially a statement about the symmetrization of the dual variety of the Segre variety. It says that the symmetrization of this dual variety becomes the union of several other varieties (with multiplicities).” I am told that for a given $n$ and $d$ this formula produces a single polynomial.

I need to know what these specific polynomials look like for the cases where $n = 2$ and $d = 2, 3, 4$. Any help would be greatly appreciated.

• This reference may be too long to help you, but the book by Gelfand, Kapranov, and Zelevinsky Discriminants, Resultants, and Multidimensional Determinants may help you. (I believe that their term multi-dimensional determinant is the same thing as a hyperdeterminant.) In particular, the book was recommended to me by a professor while giving a lecture about how the (hyper)determinant of an arbitrary tensor can be described using dual varieties. You can see him giving a related talk here: youtube.com/… Commented Oct 7, 2017 at 16:52
• The professor also mentioned Chow varieties frequently during his aforementioned lecture. And he has also done a lot of research about tensors, and in particular symmetric tensors, see e.g. here: ams.org/journals/notices/201606/rnoti-p604.pdf So I am fairly confident that the theory he was talking about is the same which you are working with now. Commented Oct 7, 2017 at 16:55