# Example Type [300b] -- An example of doubling construction

To get a quartic form $F$ of type [300b], we start with a set of $7$ points and let $F$ be power sum of them.

 i1 : kk = ZZ/101; i2 : R = kk[x_0..x_3]; i3 : HT = bettiStrataExamples(R); i4 : MGamma = (HT#"[300b]")_0 o4 = | 1 0 0 0 1 19 -8 | | 0 1 0 0 1 19 -22 | | 0 0 1 0 1 -10 -29 | | 0 0 0 1 1 -29 -24 | 4 7 o4 : Matrix R <--- R i5 : F = quartic MGamma;

We check the type of $F$.

 i6 : quarticType F o6 = [300ab]

The function quarticType cannot distinguish between type [300a] and [300b]. However, given MGamma, we now check that $F$ is of type [300b]. Let $I_{\Gamma}$ be the ideal defining the $7$ points.

 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 i9 : IGamma = pointsIdeal MGamma; o9 : Ideal of R i10 : degree IGamma o10 = 7 i11 : decompose IGamma -- 7 points, therefore the rank is at most 7 o11 = {ideal (x , x , x ), ideal (x , x , x ), ideal (x , x , x ), ideal (x , 3 2 1 3 2 0 3 1 0 2 ----------------------------------------------------------------------- x , x ), ideal (x - x , x - x , x - x ), ideal (x + 31x , x + 1 0 2 3 1 3 0 3 2 3 1 ----------------------------------------------------------------------- 32x , x + 32x ), ideal (x + 3x , x - 43x , x - 34x )} 3 0 3 2 3 1 3 0 3 o11 : List i12 : betti res IGamma 0 1 2 3 o12 = total: 1 4 6 3 0: 1 . . . 1: . 3 . . 2: . 1 6 3 o12 : BettiTally

Let $Q$ be the quadratic part of $I_{\Gamma}$. We check that $Q$ is a complete intersection. Performing Construction 2.17, we obtain a doubling of $I_{\Gamma}$, which equals $F^{\perp}$.

 i13 : Q = ideal super basis(2,IGamma); o13 : Ideal of R i14 : betti res Q 0 1 2 3 o14 = total: 1 3 3 1 0: 1 . . . 1: . 3 . . 2: . . 3 . 3: . . . 1 o14 : BettiTally i15 : Ip = Q:IGamma; o15 : Ideal of R i16 : betti res Ip 0 1 2 3 o16 = total: 1 3 3 1 0: 1 3 3 1 o16 : BettiTally i17 : v = random(2,(Fperp:Ip)); i18 : Fperp == IGamma + v*Ip o18 = true