We study moduli of ``self-associated'' sets of points in ${ \bf P}^n$ for small $n$. In particular, we show that for $n=5$ a general such set arises as a hyperplane section of the Lagrangean Grassmanian $LG(5,10) \subset { \bf P}^{ 15}$ (this was conjectured by Eisenbud-Popescu in { \it Geometry of the Gale transform}, J. Algebra 230); for $n=6$, a general such set arises as a hyperplane section of the Grassmanian $G(2,6) \subset { \bf P}^{ 14}$. We also make a conjecture for the next case $n=7$. Our results are analogues of Mukai's characterization of general canonically embedded curves in ${ \bf P}^6$ and ${ \bf P}^7$, resp.
Categories: math.AG math.AC