Families of Calabi-Yau elliptic fibrations in $\mathbb{P}({\mathcal{L}}^a\oplus {\mathcal{L}}^b\oplus {\mathcal{O}}_B)$

Let $B$ be a smooth projective surface, and $\mathcal{L}$ an ample line bundle on $B$. The aim of this parer is to study the families of elliptic Calabi–Yau threefolds sitting in the bundle $$\mathbb{P}$($\mathcal{L}$^a \oplus $\mathcal{L}$^b \oplus $\mathcal{O}$_B)$ as anticanonical divisors. We will show that the number of such families is finite.