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We consider the Hilbert scheme Hilb^{d+1}(C^d) of (d+1) points in affined-space C^d (d > 2), which includes the square of any maximal ideal. Wedescribe equations for the most symmetric affine open subscheme ofHilb^{d+1}(C^d), in terms of Schur modules. In addition we prove thatHilb^{d+1}(C^d) is reducible for n>d>11.

On the symmetric subscheme of Hilbert scheme of points