I'm trying to understand the relationship between the different models of del Pezzo surfaces of degree $2$.


Let $k$ be a field of characteristic not equal to $2$. Usually, del Pezzo surfaces of degree $2$ are considered as hypersurfaces of degree $4$ in $\mathbb{P}(1,1,1,2)$:
$$S: \quad w^2 = f(x,y,z)$$
with $\deg f = 4$.


There are however other natural models, such as hypersurfaces of bidegree $(2,2)$ in $\mathbb{P}^1 \times \mathbb{P}^2$:
$$q(s,t,x,y,z) = 0,$$
where $q$ is bihomogeneous of degree (2,2) in $(s,t)$ and $(x,y,z)$, respectively.


Call these models of type $1$ and $2$, respectively. I'm trying to understand how one goes between these different models.



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I know how one *should* go about doing this. The anticanonial class $-K\_S$ is the fibre of the projection onto $\mathbb{P}^2$. So a basis for the anticanonical sections is $x,y,z$. Now the space $H^0(S, -2K\_S)$ is $7$-dimensional, so there should be a section $w$ in here which is linearly indenpendent from the obvious sections $x^2,xy,xz,y^2,yz,z^2$. With this in hand, we obtain the required map
$$S \to \mathbb{P}(1,1,1,2), \quad (s,t) \times (x,y,z) \mapsto (x,y,z,w).$$
But what is $w$? I don't see how to determine $w$ explicitly.



Del Pezzo surfaces of degree $2$
One can transform birationally $S \subset \mathbb{P}^1 \times \mathbb{P}^2$ to $S' \subset\mathbb{P}(1,1,1,2)$ as follows. Choose a point $P \in \mathbb{P}^1$ such that the conic 
$$
C := (P \times \mathbb{P}^2) \cap S
$$
is smooth. Blow up $P \times C \subset \mathbb{P}^1 \times \mathbb{P}^2$. Then blow down the proper preimage of $\mathbb{P}^1 \times C$. Then you will get $\mathbb{P}\_{\mathbb{P}^2}(O \oplus O(-2))$. The strict transform of $P \times \mathbb{P}^2$ is the exceptional section and it does not intersect the strict transform of $S$. If you contract it, you get $\mathbb{P}(1,1,1,2)$ with an embedded image of $S$.


