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 |