## Classification of elliptic and K3 fibrations birational to some $\QQ$-Fano 3-folds

J. Math. Sci. Univ. Tokyo
Vol. 13 (2006), No. 1, Page 13--42.

Ryder, Daniel
Classification of elliptic and K3 fibrations birational to some $\QQ$-Fano 3-folds
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.