# Anticanonical line bundle of a threefold in a product of a hypersurface and P1

algebraic-geometrysheaf-theory

I am reading a paper and there I have the following:

$$F:=Y\times \mathbb P^1$$ where $$Y$$ is a sextic hypersurface in the weighted projective space $$\mathbb P(1^3,2,3)$$.

(1) Let $$X\subset F$$ be a Fano threefold that is a (1,1)-section of
$$F$$.

The anticanonical line bundle

(2) $$\omega_X^{\vee}\cong (\mathcal O_F(2,2)\otimes \mathcal O_F(-X))|_X\cong \mathcal O_F(1,1)|_X$$

The conormal sequence for the inclusion $$X\hookrightarrow F$$ twisted by the anticanonical line bundle.

(3) $$0\rightarrow \mathcal O_X\rightarrow\Omega_F^1(1,1)|_X\rightarrow \Omega_X^1(1,1)\rightarrow 0$$

In (1), what does "$$X$$ is a (1,1)-section of $$F$$" mean?

In (2), How can I prove the last isomorphism? I know that the anticanonical line bundle is $$\omega_X^{\vee}\cong \omega_F^{\vee}\otimes I/I^2\cong \mathcal O_F(2,2)\otimes \mathcal O_F(-X))$$, since $$\omega_F^{\vee}=\pi_1^*\mathcal O(6-(3+2+3))|_Y\otimes \pi_2^*\mathcal O(-2)=\mathcal O_F(2,2)$$ according with the notation of the paper. Also I know that $$I/I^2=\mathcal O_F(-X)$$ but I don't understand how to prove the last isomorphism.

In (3), I know that the the conormal sequence for the inclusion $$X\hookrightarrow F$$ is $$0\rightarrow I/I^2\rightarrow\Omega_F^1|_X\rightarrow \Omega_X^1\rightarrow 0$$ but, why after twist it by $$\omega_X^{\vee}$$ the firts sheaf is $$\mathcal O_X$$? I can't conclude that from (2).

I guess $$\mathcal{O}_Y(1)$$ denotes the ample generator of the Picard group of $$Y$$ --- this is the restriction of the reflexive sheaf $$\mathcal{O}(1)$$ from the weighted projective space. Note that by adjunction formula $$\omega_Y \cong \mathcal{O}_Y(-2).$$ Further, I think $$X \subset F = Y \times \mathbb{P}^1$$ is defined as the hypersurface from the linear system $$\mathcal{O}_F(1,1) = \mathcal{O}_Y(1) \boxtimes \mathcal{O}_{\mathbb{P}^1}(1)$$ (this answers (1)).
Furthermore, (2) follows from the standard isomorphism $$I/I^2 \cong \mathcal{O}_F(-1,-1)\vert_X = \mathcal{O}_X(-1,-1)$$ and (3) is obtained be tensoring the conormal sequnce with $$\mathcal{O}_X(1,1)$$.