TL;DR
This paper constructs examples in positive characteristic where the minimal model program (MMP) fails to extend from the special fiber to the entire family, highlighting limitations of MMP in certain algebraic settings.
Contribution
It provides explicit examples showing that the MMP does not always extend in families over a DVR in positive characteristic, revealing new limitations of the MMP.
Findings
MMP can fail to extend from fibers to families in positive characteristic.
Constructed examples with mild singularities demonstrate this failure.
Highlights the need for caution in applying MMP in positive characteristic contexts.
Abstract
We construct examples of families of pairs over a DVR of positive characteristic, with very mild singularities, such that the MMP on the closed fiber does not extend to a relative MMP.
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Taxonomy
TopicsHolomorphic and Operator Theory · Algebraic structures and combinatorial models · Geometric and Algebraic Topology