We construct via linkage an arithmetically Gorenstein 3-fold $X = X_7& \cup X_{4} \cup X_{4}' \subset \bf{P}^7$, of degree 15, with components of degrees $7, 4, 4$, having Betti table of type [562]. For an artinian reduction $A_F$, the ideal $F^\perp$ contains a pencil of ideals $I_\Gamma$, where $\Gamma=\Gamma_3\cup p_1\cup p_2$, the union of a three points in a fixed line and two independant points outside the line. So we construct $X_7$ in the intersection of two cubics in a P5 and $X_4$ and $X_4'$ as quartics in an independant P4s. In the construction the intersection $X\cup (X'\cap X'')$ of a component $X$ with the two others is an anticanonical divisor on $X$.
The betti table is $\phantom{WWWW} \begin{matrix} &0&1&2&3&4\\ \text{total:}&1&9&16&9&1\\ \text{0:}&1&\text{.}&\text{.}&\text{.}&\text{.}\\ \text{1:}&\text{.}&5&6&2&\text{.}\\ \text{2:}&\text{.}&2&4&2&\text{.}\\ \text{3:}&\text{.}&2&6&5&\text{.}\\ \text{4:}&\text{.}&\text{.}&\text{.}&\text{.}&1\\ \end{matrix} $
$X_7$ is linked to a reducible quadric 3-fold $Y$ in a complete intersection $(1,1,3,3)$. $X_4$ and $X_4'$ are quartic 3-folds that each intersect $X_7$ in a cubic surface, while they intersect each other in a plane. The cubic surfaces are the intersection of $X_7$ with the components of $Y$, and the plane is the intersection of these components.
i1 : kk=QQ;
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i2 : U=kk[y0,y1,y2,y3,y4,y5,y6,y7];
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i3 : P5=ideal(y0,y1);--a P5
o3 : Ideal of U
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i4 : P3a=ideal(y0,y1,y2,y3);-- a P3
o4 : Ideal of U
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i5 : P3b=ideal(y0,y1,y2,y4);-- another P3
o5 : Ideal of U
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i6 : P4a=ideal(y0,y2,y3);-- a P4
o6 : Ideal of U
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i7 : P4b=ideal(y1,y2,y4);-- a P4
o7 : Ideal of U
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i8 : X2=ideal(y0,y1,y2,y3*y4);--a reducible quadric
o8 : Ideal of U
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i9 : CI1133=P5+ideal(random(3,X2),random(3,X2));--a complete intersection (1,1,3,3) that contain X2.
o9 : Ideal of U
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i10 : X7=CI1133:X2; -- a 3-fold of degree 7, linked (1,1,3,3) to X2
o10 : Ideal of U
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