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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Possibility 1:

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Possibility 2:

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Possibility 3:

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Possibility 4:

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