Hilbert Functions of symbolic powers of ideals of points on plane conic.
Let Z be a simple point subscheme of P2 supported at any points of a plane conic. Let I=I(Z) be the ideal, and consider I(m).
There are four possibilities: (1) n points on an irreducible conic;
(2) n=n1+n2 points with n1 >= n2 >= 2, where n1 points are on one line,
n2 on another and none are where the two lines meet; (3)
n=n1+n2+1 points with n1 >= n2 >= 2, where n1 points are on one line,
n2 on another and one point where the two lines meet; and (4) n+1 points
with n > 1 on a line and one point off the line.
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