English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The main results of this paper are already known (V.V. Shokurov, thenon-vanishing theorem, 1985). Moreover, the non-$\mathbb{Q}$-factorial MMP wasmore recently considered by O~Fujino, in the case of toric varieties(Equivariant completions of toric contraction morphisms, 2006), for klt pairs(Special termination and reduction to pl flips, 2007) and more generally forlog-canonical pairs (Foundation of the minimal model program, 2014). Here werewrite the proofs of some of these results, by following the proofs given byY. Kawamata, K. Matsuda, and K. Matsuki (Introduction to the minimal modelproblem, 1985) of the same results in $\mathbb{Q}$-factorial MMP. And, in thefamily of $\mathbb{Q}$-Gorenstein spherical varieties, we answer positively tothe questions of existence of flips and of finiteness of sequences of flips. Iapologize for the first version of this paper, which I wrote without knowingthat these results already exist.

A Minimal Model Program for $\mathbb{Q}$-Gorenstein varieties