# Type [562] with lifting of type I, a CY of degree 15 via linkage -- lifting to a 3-fold with components of degree 8, 7

We construct via linkage an arithmetically Gorenstein 3-fold $X = X_{8} \cup X_{7} \subset \bf{P}^7$, of degree 15, with two components of degrees 7 and 8, having Betti table of type [562]. For an artinian reduction $A_F$, the ideal $F^\perp$ contains a pencil of ideals $I_\Gamma$, where $\Gamma=\Gamma_3\cup\Gamma_2$, the union of a three points in a line $L$ and two fixed points on a line $L'$ skew to $L$ . So we construct $X_8$ in the complete intersection of two cubics in a P5 and $X_7$ in a complete intersection $(2,4)$ in another P5. In the construction the intersection $X\cup X'$ of a component $X$ with the other 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 a 3-fold of degree 7 linked via a $(2,4)$ complete intersection to a $\PP^3$ in a $\PP^5$. The other component $X_8$ of $X$ is a 3-fold of degree 8 linked via a $(3,3)$ complete intersection to another $\PP^3$ in another $\PP^5$. These are constructed do that $X_7$ and $X_8$ intersect in a quartic surface $Z_4$ in the $\PP^3$ which is the intersection of the span of $X_7$ and $X_8$.

 i1 : kk=QQ; i2 : U=kk[y0,y1,y2,y3,y4,y5,y6,y7]; i3 : P5c=ideal(y0,y1); -- a P5 o3 : Ideal of U i4 : P5a=ideal(y2,y3); --P5 of S8 o4 : Ideal of U i5 : P5b=ideal(y4,y5); --P5 of S7 o5 : Ideal of U i6 : P3ac=P5a+P5c;-- P3 intersection of P5a and P5c o6 : Ideal of U i7 : P3bc=P5b+P5c; o7 : Ideal of U i8 : P1=P5a+P5b+P5c; --a line L, the intersection of all three P5s o8 : Ideal of U i9 : F=matrix{{y0,random(2,U),random(2,P1)},{y1,random(2,U),random(2,P1)}}; 2 3 o9 : Matrix U <--- U
 i10 : X8=P5a+minors(2,F);-- a 3-fold of degree 8 in P5a o10 : Ideal of U i11 : Z4=P5c+X8; -- a quartic surface in P3ac that contains L o11 : Ideal of U i12 : XY=P5c+ideal(random(2,intersect(P5a,P5b)),random(4,intersect(Z4,P5b))); o12 : Ideal of U
 i13 : X7=XY:P3bc;--a 3-fold of degree 7 in P5b o13 : Ideal of U i14 : (dim X7, degree X7) o14 = (4, 7) o14 : Sequence i15 : betti res X7 0 1 2 3 4 o15 = total: 1 5 9 7 2 0: 1 2 1 . . 1: . 1 2 1 . 2: . . . . . 3: . 2 6 6 2 o15 : BettiTally
 i16 : X15=intersect(X7,X8);--a 3-fold of degree 15 in P7 with betti table of type [562], the union of X7 and X8. o16 : Ideal of U i17 : (dim X15, degree X15) o17 = (4, 15) o17 : Sequence i18 : betti res X15 0 1 2 3 4 o18 = total: 1 9 16 9 1 0: 1 . . . . 1: . 5 6 2 . 2: . 2 4 2 . 3: . 2 6 5 . 4: . . . . 1 o18 : BettiTally