RNCinP2Notes.txt

We are interested in the case of the matroid of general changes of
coordinate of the rational normal curve V(xz-y^2).  This is a general
quadric in P^2, so we can assume that I = <ax^2+bxy+cxz+dy^2+eyz+fz^2>.

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d=3

For I_3, the 3 x 3 minors of the Macaulay matrix vanish when
3 of the monomials are not divisible by x (12 total).

The 3 x 3 submatrix of the Macaulay matrix then has a row of zeros.

To check: non-vanishing.

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d=4
For I_4, the 6 x 6 minors of the Macaulay matrix seem to vanish when
we are in one of the three situations happen:

a) All monomials  are not divisible by x^4, x^3, or x^2
(we think 252 = binom{9}{6}*3 up to symmetry)

b) 4 monomials are not divisible by x, plus any two others
(we think 605 not in the previous case, up to symmetry)

c) All monomials are not divisible by x or not divisible by y 
(48 not in previous cases no in symmetry)

d) The monomials with x exponent 1 or 3 (3 up to symmetry)

e) {x^3y, x^3z, y^3x, y^3z, z^3x, z^3y}  (unique up to symmetry).

This is a total of 907

This agrees with the computation.

To check: non-computer proof

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