# Example Type [300c] -- The third family of type [300]

To get a quartic form $F$ of type [300c], we start with a set of $7$ points, with $3$ of them in a line, and let $F$ be their power sum.

 i1 : kk = ZZ/101; i2 : R = kk[x_0..x_3]; i3 : HT = bettiStrataExamples(R); i4 : MGamma = (HT#"[300c]")_0 o4 = | 1 0 1 -38 34 -18 -28 | | 0 1 1 -16 19 -13 -47 | | 0 0 0 39 -47 -43 38 | | 0 0 0 21 -39 -15 2 | 4 7 o4 : Matrix R <--- R i5 : IGamma = pointsIdeal MGamma; o5 : Ideal of R i6 : F = quartic MGamma;

We check the type of $F^{\perp}$ and see that the quadratic part $Q$ of $F^{\perp}$ is not a complete intersection.

 i7 : quarticType F o7 = [300c]
 i8 : Fperp = inverseSystem F; o8 : Ideal of R i9 : betti res Fperp 0 1 2 3 4 o9 = total: 1 7 12 7 1 0: 1 . . . . 1: . 3 . . . 2: . 4 12 4 . 3: . . . 3 . 4: . . . . 1 o9 : BettiTally i10 : Q = ideal super basis (2,Fperp); o10 : Ideal of R i11 : betti res Q 0 1 2 3 o11 = total: 1 3 4 2 0: 1 . . . 1: . 3 . . 2: . . 4 2 o11 : BettiTally

Now we construct a doubling of $I_{\Gamma}$, which is not necessary the same as $F^{\perp}$, but is of type [300c].

Let $J$ be a subideal of $I_{\Gamma}$ which is a $(2,2,3)$ complete intersection.

 i12 : J = ideal(random(2,IGamma),random(2,IGamma),random(3,IGamma)); o12 : Ideal of R i13 : betti res J 0 1 2 3 o13 = total: 1 3 3 1 0: 1 . . . 1: . 2 . . 2: . 1 1 . 3: . . 2 . 4: . . . 1 o13 : BettiTally

The colon ideal $I_{p}=J:I_{\Gamma}$ is a set of $5$ points. Performing Construction 2.17, we can find a doubling of $I_{\Gamma}$, which is of type [300c].

 i14 : Ip = J : IGamma 2 2 2 o14 = ideal (x x - 18x + 14x x + 49x x - 39x x + 44x , x x + 21x - 1 2 2 0 3 1 3 2 3 3 0 2 2 ----------------------------------------------------------------------- 2 2 2 43x x - 50x x + 14x x + 35x , x - 42x - 29x x - 32x x - 36x x + 0 3 1 3 2 3 3 1 2 0 3 1 3 2 3 ----------------------------------------------------------------------- 2 2 2 2 2 14x , x x + 19x + 44x x - 36x x - 23x x - 35x , x + 2x - 23x x 3 0 1 2 0 3 1 3 2 3 3 0 2 0 3 ----------------------------------------------------------------------- 2 - 32x x - 18x x - 31x ) 1 3 2 3 3 o14 : Ideal of R i15 : betti res (Fperp:Ip) 0 1 2 3 4 o15 = total: 1 7 11 8 3 0: 1 . . . . 1: . 7 8 . . 2: . . 3 8 3 o15 : BettiTally i16 : l = random(1,R); i17 : betti res (IGamma+l*Ip) 0 1 2 3 4 o17 = total: 1 7 12 7 1 0: 1 . . . . 1: . 3 . . . 2: . 4 12 4 . 3: . . . 3 . 4: . . . . 1 o17 : BettiTally