1) This is the Chow variety of degree $n$ zero cycles in $\mathbb{P}^{d-1}$.

2) Yes, this collection of polynomials can be bundled together into the Brill form or covariant.

3) Rather explicit descriptions of the Brill equations can be found in the book by Gelfand, Kapranov and Zelevinsky on resultants. There is also a paper by Rota and Stein. But first check out Emmanuel Briand's page and in particular the articles "Covariants decomposing on totally decomposable forms" and "Brill's equations for the subvariety of factorizable forms" and if you read French (or German) the translation of the original article by Gordan (respectively the article itself).

As an aside, analogues of the Brill equations for the variety of forms which are powers of forms of degree dividing $n$ have been given recently in my paper with Chipalkatti "On Hilbert covariants".

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1) This is the Chow variety of degree $n$ zero cycles in $\mathbb{P}^{d-1}$.

2) Yes, this collection of polynomials can be bundled together into the Brill form or covariant.

3) Rather explicit descriptions of the Brill equations can be found in the book by Gelfand, Kapranov and Zelevinsky on resultants. There is also a paper by Rota and Stein. But first check out Emmanuel Briand's page and in particular the articles "Covariants decomposing on totally decomposable forms" and "Brill's equations for the subvariety of factorizable forms" and if you read French (or German) the translation of the original article by Gordan (respectively the article itself).