To get a quartic form $F$ of type [300a], we start with a point set $\Gamma$ which is a complete intersection of three quadric forms. Then we let $F$ be a general element in the space spanned by $v_{4}(\Gamma)\subset\mathbb{P}^{34}$.
i1 : kk = ZZ/101;
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i2 : R = kk[x_0..x_3];
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i3 : HT = bettiStrataExamples(R);
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i4 : MGamma = (HT#"[300a]")_0
o4 = | 1 1 1 1 1 1 1 1 |
| 2 2 2 2 -2 -2 -2 -2 |
| 3 3 -3 -3 3 3 -3 -3 |
| 1 -1 1 -1 1 -1 1 -1 |
4 8
o4 : Matrix R <-- R
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i5 : linforms = flatten entries((vars R) * MGamma);
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i6 : F = sum for ell in linforms list random(kk)*ell^4
4 3 2 2 3 4 3 2 2
o6 = - 11x + 36x x + 39x x + 43x x + 26x + 12x x + 22x x x + 43x x x
0 0 1 0 1 0 1 1 0 2 0 1 2 0 1 2
------------------------------------------------------------------------
3 2 2 2 2 2 3 3 4 3
- 38x x + 12x x - 38x x x + 48x x + 7x x - 35x x + 18x - 3x x -
1 2 0 2 0 1 2 1 2 0 2 1 2 2 0 3
------------------------------------------------------------------------
2 2 3 2 2
35x x x - 36x x x - 13x x + 14x x x + 29x x x x - 45x x x +
0 1 3 0 1 3 1 3 0 2 3 0 1 2 3 1 2 3
------------------------------------------------------------------------
2 2 3 2 2 2 2 2 2
20x x x - 12x x x + 42x x + 35x x + 7x x x + 39x x + 36x x x +
0 2 3 1 2 3 2 3 0 3 0 1 3 1 3 0 2 3
------------------------------------------------------------------------
2 2 2 3 3 3 4
22x x x + 12x x - 3x x + 22x x - 29x x - 11x
1 2 3 2 3 0 3 1 3 2 3 3
o6 : R
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We check the Betti table of $F^\perp$.
i7 : Fperp = inverseSystem F;
o7 : Ideal of R
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i8 : betti res Fperp
0 1 2 3 4
o8 = total: 1 7 12 7 1
0: 1 . . . .
1: . 3 . . .
2: . 4 12 4 .
3: . . . 3 .
4: . . . . 1
o8 : BettiTally
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Let $Q$ be the quadratic part of $F^{\perp}$. We check that $Q=I_{\Gamma}$.
i9 : Q = ideal super basis(2,Fperp);
o9 : Ideal of R
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i10 : Q == pointsIdeal(MGamma)
o10 = true
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We know that $\Gamma$ is a minimal apolar set to $F$. The doubling of $I_{\Gamma}$ is always a complete intersection. Therefore, $F^{\perp}$ cannot be obtained as a doubling of $I_{\Gamma}$ in this case.