We construct via linkage an arithmetically Gorenstein irreducible 3-fold $X = X_{16} \bf{P}^7$, of degree 16, having Betti table of type [420]. For an artinian reduction $A_F$, the quadratic part of the ideal $F^\perp$ is the ideal of six points on a twisted cubic curve. We construct $X_{16}$ as an anticanonical divisor in the fourfold intersection of a cubic scroll and quadric.
The betti table is $\phantom{WWWW} \begin{matrix} &0&1&2&3&4\\ \text{total:}&1&6&10&6&1\\ \text{0:}&1&\text{.}&\text{.}&\text{.}&\text{.}\\ \text{1:}&\text{.}&4&2&\text{.}&\text{.}\\ \text{2:}&\text{.}&2&6&2&\text{.}\\ \text{3:}&\text{.}&\text{.}&2&4&\text{.}\\ \text{4:}&\text{.}&\text{.}&\text{.}&\text{.}&1\\ \end{matrix} $
i1 : kk=QQ; |
i2 : U=kk[y0,y1,y2,y3,y4,y5,y6,y7]; |
i3 : P4=ideal(y0,y1,y2);--a P4 o3 : Ideal of U |
i4 : T3=minors(2,matrix{{y0,y1,y2},{y3,y4,y5}});--a cubic 5-fold scroll with P4 as a ruling
o4 : Ideal of U
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i5 : X2=P4+ideal(random(2,U));-- a quadric 3-fold in T3 o5 : Ideal of U |
i6 : X18=T3+ideal(random(2,X2),random(3,X2));--a 3-fold of degree 18 in T3 that contains X2 o6 : Ideal of U |
i7 : X16=X18:X2;--a 3-fold of degree 16 in T3 with betti table of type 420 o7 : Ideal of U |
i8 : (dim X16, degree X16) o8 = (4, 16) o8 : Sequence |
i9 : betti res X16
0 1 2 3 4
o9 = total: 1 6 10 6 1
0: 1 . . . .
1: . 4 2 . .
2: . 2 6 2 .
3: . . 2 4 .
4: . . . . 1
o9 : BettiTally
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