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# 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

## See also

- [QQ] -- Quaternary Quartic Forms and Gorenstein rings (Kapustka, Kapustka, Ranestad, Schenck, Stillman, Yuan, 2021)

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/QuaternaryQuartics/Section2Doc.m2:212:0.