Equigenerated Gorenstein ideals of codimension three.

One focus on the structure of a codimension $3$ homogeneous Gorenstein ideal $I$ in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. This degree is tied to the minimal number of generators of $I$ and the degree of the entries of an associated skew-symmetric matrix and one shows, conversely, in a characteristic-free way, that any such data are fulfilled by some Gorenstein ideal. It is proved that a codimension $3$ homogeneous Gorenstein ideal $I\subset \kk[x,y,z]$ cannot be generated by general forms of fixed degree, except precisely when $d=2$ or the ideal is a complete intersection. The question as to when the link $(\ell_1^m,\ldots,\ell_n^m):\mathfrak{f}$ is equigenerated, where $\ell_1,\ldots,\ell_n$ are independent linear forms and $\mathfrak{f}$ is a form, is given a solution in some important cases.