# Example Type [300a] -- An example of an apolar ideal of a quartic that cannot be obtained as a doubling of it's apolar set

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; i2 : R = kk[x_0..x_3]; i3 : HT = bettiStrataExamples(R); 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 i5 : linforms = flatten entries((vars R) * MGamma); 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

We check the Betti table of $F^\perp$.

 i7 : Fperp = inverseSystem F; o7 : Ideal of R 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

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 i10 : Q == pointsIdeal(MGamma) o10 = true

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.