Existence of Mori fibre spaces for 3-folds in char p

A note on the fibres of Mori fibre spaces

“…[HX15], [Bir16], [BW17]), the picture that we delineated in the Introduction holds almost in the same way in characteristic > 5 as long as we only focus in dimension 2 and 3. Hence, it is natural to try to extend the results in [CFST16] to positive characteristic.…”

Abundance for non‐uniruled 3‐folds with non‐trivial Albanese maps in positive characteristics

“…Proof For (1) please refer to [, Theorem 1.2 and Proposition 7.3]. For (2), refer to . For (3.1), please refer to [, , ].…”

“…For (2), refer to . For (3.1), please refer to [, , ]. For (3.2) and (3.3) please refer to [, Theorem 1.5 and 1.6 and the remark below 1.6].…”

“…And (3.2) also can be obtained from . Assertion (3.4) follows from the cone theorem [, Theorem 1.1]. Indeed, otherwise we can find an extremal ray generated by a rational curve , so is contained in a fiber of since contains no rational curves, this contradicts that is ‐nef.…”

“…For (4), is obviously pseudo‐effective because otherwise, will be ruled by horizontal (with respect to ) rational curves by (2), which contradicts that is non‐uniruled. The exceptional locus of a flip contraction is of dimension 1, so it does not intersect , neither does that of an extremal divisorial contraction because it is uniruled (see the proof of [, Lemma 3.2]). Running an LMMP for , by induction we get a needed map .…”