Fiberwise compactification of a LG model It is believed that a mirror of $\mathbb{CP}^2$ is a fiberwise compactification of the family $$W \colon (\mathbb{C}^\times)^2 \rightarrow \mathbb{C}, \quad (x,y)\mapsto x+y+\frac{1}{xy}.$$ Is it obtained by considering a pencil $\mathbb{CP}^2 \rightarrow \mathbb{CP}^1$
$$
W_1:[X:Y:Z]\mapsto[X^2Y+XY^2+Z^3:XYZ], 
$$
blowing-up the base locus of $W_1$ and removing the fiber of $0$ and $\infty$? Is it true that a fiberwise compactification is always obtained in this way and it is canonical in a certain sense? 
 A: For the mirror of $\mathbb{CP}^2$, a smooth fiber of $W$ is a 3-punctured torus, and you can compactify the fibers by filling in 1, 2 or 3 punctures. This corresponds, in $\mathbb{CP}^2$, to different divisors.
The original LG model you wrote down is a toric mirror of $\mathbb{CP}^2$, which means it's mirror to the toric divisor $D_0$. So after removing it, you get $(\mathbb{C}^\ast)^2$. In this case, mirror symmetry gives you an equivalence
$D^\pi\mathscr{W}(T^\ast T^2)\cong D^b\mathit{Coh}((\mathbb{C}^\ast)^2)$,
which can be obtained from the triangulated equivalence
$D\mathscr{F}(T^\ast T^2,W)\cong D^b\mathit{Coh}(\mathbb{CP}^2)$
by applying a categorical version of localization, where the left-hand side is the Fukaya-Seidel category.
Filling in one of the punctures compactifies $(\mathbb{C}^\ast)^2$ to the affine algebraic surface $M=\mathit{Spec}(\mathbb{C}[x,y]/(xy-1))$, which is in fact self-mirror, i.e. $M\cong M^\vee$. This is to say, in this case the corresponding divisor $D_1\subset\mathbb{CP}^2$ is the union of a line and a conic. By a theorem of Seidel, we have an equivalence
$D\mathscr{F}(M,W)\cong D\mathscr{F}(T^\ast T^2,W),$
which in fact follows from the fact that in this case we can choose a trivial bulk term to obtain a trivial deformation of the directed $A_\infty$ category associated to the fiberwise compactification of the Lefschetz fibration. This result holds also in the cases when filling in 2 or 3 points in a smooth fiber. From this we see that these LG models are all mirrors of $\mathbb{CP}^2$.
In the case when 2 points are filled in for each smooth fiber of $W$, the divisor $D_2\subset\mathbb{CP}^2$ is a nodal elliptic curve, while in the case when all the 3 punctures are filled in, $D_3$ is a smooth elliptic curve.
A large class of fiberwise compactifications of Lefschetz fibrations arises in the way you have just described, see the paper of Seidel (https://arxiv.org/pdf/1504.06317v2.pdf) for the general set up. However, it's not true for all fiberwise compactifications, simply because there are Lefschetz fibrations which do not come from Lefschetz pencils.
