Configurations of eigenpoints

This note is motivated by Question 16 of http://cubics.wikidot.com (see also [16]): Which configurations of 15 points in P3 arise as eigenpoints of a cubic surface? We prove that a general eigenscheme in Pn is the complete intersection of two suitable smooth determinantal curves on a smooth determinantal surface. Moreover, we prove that the converse result holds if n = 3, providing an answer in any degree to the cited question. Finally, we show that any general set of points in P3 can be enlarged to an eigenscheme of a partially symmetric tensor.