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

Abstract

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.

arXiv comment15 pages, 2 figures. Revised version: main result uneffected, found and fixed a mistake in the previous proof. Added a new section with examples