A complete classification is presented of elliptic and K3 fibrations birational to certain mildly singular complex Fano 3-folds. Detailed proofs are given for one example case, namely that of a general hypersurface $X$ of degree 30 in weighted $\PP^4$ with weights 1,4,5,6,15; but our methods apply more generally. For constructing birational maps from $X$ to elliptic and K3 fibrations we use Kawamata blowups and Mori theory to compute anticanonical rings; to exclude other possible fibrations we make a close examination of the strictly canonical singularities of $\XnH$, where $\HH$ is the linear system associated to the putative birational map and $n$ is its anticanonical degree.
|Translated title of the contribution||Classification of elliptic and K3 fibrations birational to some Q-Fano 3-folds|
|Pages (from-to)||13 - 42|
|Number of pages||20|
|Journal||Journal of Mathematical Sciences (University of Tokyo)|
|Publication status||Published - Jan 2006|