Nonrational del Pezzo fibrations

Abstract

Let $X$ be a general divisor in $|3M+nL|$ on the rational scroll $\mathrm{Proj}({\oplus }_{i=1}^4{\mathcal{O}}_{{\mathbb{P}}^1}(d_i))$, where $d_i$ and $n$ are integers, $M$ is the tautological line bundle, $L$ is a fibre of the natural projection to ${\mathbb{P}}^1$, and $d_1⩾...⩾d_4=0$. We prove that $X$ is rational $⟺$ $d_1=0$ and $n=1$.

8 pages, short version, to appear in Advances in Geometry

References in corpus (2)

• Non-rationality of a three-dimensional Fano variety of index 2 and degree 1
• On factoriality of nodal threefolds