  We study the 0-dimensional schemes supported at one point in $n$-space which
are $m$-symmetric, i.e. they intersect any curves thru the point with length
$m$. We show that the maximal length for such a scheme is $m^n$ ($m$-superfat
points) and we study properties of such schemes, in particular for $n=2$. We
also study varieties defined by such schemes on Veronese and Segre-veronese
varieties.
