Nonrational del Pezzo fibrations

Details Nonrational del Pezzo fibrations Nonrational del Pezzo fibrations

2004-07-20

arxiv.org/abs/math/0407343v4

Mathematics

Algebraic Geometry

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

Scientific paper

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 $d_{1}\geqslant...\geqslant d_{4}=0$. We prove that $X$ is rational $\iff$ $d_{1}=0$ and $n=1$.

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