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

Superfat points and associated tensors