# Type  with a lifting of type II, a CY of degree 15 via linkage -- lifting to a 3-fold with components of degrees 7, 4, 4

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 . 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; i2 : U=kk[y0,y1,y2,y3,y4,y5,y6,y7]; i3 : P5=ideal(y0,y1);--a P5 o3 : Ideal of U i4 : P3a=ideal(y0,y1,y2,y3);-- a P3 o4 : Ideal of U i5 : P3b=ideal(y0,y1,y2,y4);-- another P3 o5 : Ideal of U i6 : P4a=ideal(y0,y2,y3);-- a P4 o6 : Ideal of U i7 : P4b=ideal(y1,y2,y4);-- a P4 o7 : Ideal of U i8 : X2=ideal(y0,y1,y2,y3*y4);--a reducible quadric o8 : Ideal of U i9 : CI1133=P5+ideal(random(3,X2),random(3,X2));--a complete intersection (1,1,3,3) that contain X2. o9 : Ideal of U i10 : X7=CI1133:X2; -- a 3-fold of degree 7, linked (1,1,3,3) to X2 o10 : Ideal of U
 i11 : (dim X7, degree X7) o11 = (4, 7) o11 : Sequence
 i12 : Z3a=P3a+X7; -- a cubic surface o12 : Ideal of U i13 : Z4a=intersect(Z3a,P3b);-- the union of Z3a and P3b o13 : Ideal of U i14 : X4a=P4a+ideal(random(4,Z4a));-- a quartic 3-fold in P4 that contains a plane in P3b and the cubic surface Z3 o14 : Ideal of U
 i15 : (dim X4a, degree X4a) o15 = (4, 4) o15 : Sequence
 i16 : Z3b=X7+P3b;-- a cubic surface o16 : Ideal of U i17 : Z4b=intersect(Z3b,P3a);-- the union of Z3b and P3b o17 : Ideal of U i18 : X4b=P4b+ideal(random(4,Z4b));-- a quartic 3-fold in P4 that contains a plane in P3b and the cubic surface Z3 o18 : Ideal of U
 i19 : (dim X4b, degree X4b) o19 = (4, 4) o19 : Sequence

The union $X = X_7 \cup X_4 \cup X_4' \subset \PP^7$ has Betti table .

 i20 : X15=intersect(X7,X4a,X4b);--a 3-fold of degree 15, with betti table of type , with three components, X7 of degree 7, and X4a and X4b of degree 4. o20 : Ideal of U i21 : (dim X15, degree X15) o21 = (4, 15) o21 : Sequence i22 : betti res X15 0 1 2 3 4 o22 = total: 1 9 16 9 1 0: 1 . . . . 1: . 5 6 2 . 2: . 2 4 2 . 3: . 2 6 5 . 4: . . . . 1 o22 : BettiTally