We construct via linkage an arithmetically Gorenstein 3-fold $X = X_{11} \cup X_1 \cup X_6 \subset \bf{P}^7$, of degree 17, having Betti table of type [210]: For an artinian reduction $A_F$, the quadratic part of the ideal $F^\perp$ is the ideal of a plane and a line, and $F^\perp$ contains pencils of ideals of five points on a conic in the plane and three points on the line, and a 3-dimensional family of ideals of six points in the plane and two points on the line. In particular, $F^\perp$ contains the ideal of the intersection point of the line and the plane in addition to five points in the plane and two points on the line.
We construct $X_{11}$ in a quadric in a P6, $X_{6}$ in a quadric in a P5 and $X_{1}$ in the P4 of intersection of P5 and P6. In the construction the intersection $X\cup X'$ of a component $X$ with the other is an anticanonical divisor on $X$.
$\phantom{WWWW} \begin{matrix} &0&1&2&3&4\\\text{total:}&1&11&20&11&1\\ \text{0:}&1&\text{.}&\text{.}&\text{.}&\text{.}\\ \text{1:}&\text{.}&2&1&\text{.}&\text{.}\\ \text{2:}&\text{.}&9&18&9&\text{.}\\ \text{3:}&\text{.}&\text{.}&1&2&\text{.}\\ \text{4:}&\text{.}&\text{.}&\text{.}&\text{.}&1\\ \end{matrix} $
i1 : kk = QQ;
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i2 : U=kk[y0,y1,y2,y3,y4,y5,y6,y7];
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i3 : X1 = ideal(y0,y5,y6,y7); -- the component X1, a P3
o3 : Ideal of U
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i4 : P3a = ideal(y0,y1,y2,y3); --another P3
o4 : Ideal of U
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i5 : P4 = ideal(y0,y6,y7); --a P4
o5 : Ideal of U
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i6 : P5 = ideal(y6,y7); --a P5
o6 : Ideal of U
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i7 : P6 = ideal(y0); -- a P6
o7 : Ideal of U
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Compute the ideal of the union of $CI_{22}$ and the other $\PP^3$. Then compute a complete intersection of type $(1,2,2,2)$ which contains $\PP^3_a$ and $CI_{22}$.
i16 : Z6 = P4 + QX7 + ideal(random(3,X7Z2));--a complete intersection 2,3 in P4
o16 : Ideal of U
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i17 : X7Z6 = intersect(X7,Z6); --the union of X7 and the complete intersection 2,3 in P4
o17 : Ideal of U
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i18 : CI1233 = P6 + QX7 + ideal(random(3,X7Z6),random(3,X7Z6));--complete intersection 1233 containing Z2a and a hyperplane section of X7
o18 : Ideal of U
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i19 : X11 = CI1233:X7; --a 3-fold of degree 11.
o19 : Ideal of U
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i20 : Z4 = Z6:Z2a;--a Del Pezzo surface in P4
o20 : Ideal of U
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i21 : Y2 = P5+ideal(random(2,Z4));--a quadric 4-fold in P5 that contain Z4
o21 : Ideal of U
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i22 : Z2b = X1+Y2;-- another quadric surface in X1
o22 : Ideal of U
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i23 : Z6b = intersect(Z4,Z2b);-- a complete intersection 2,3 different from Z6a, the union of Z4 and Z2b
o23 : Ideal of U
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i24 : Y3 = P5 + ideal(random(3,Z6b));--a cubic 4-fold in P5 that contain Z6b
o24 : Ideal of U
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i25 : X6 = Y2 + Y3;
o25 : Ideal of U
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i26 : X18 = intersect(X11,X6,X1);--a AG 3-fold of degree 18, with betti table of type 210.
o26 : Ideal of U
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