MathDB

Relative non-nef loci conjecture

For a projective morphism f ⁣:X→Sf\colon X\to S and a relatively pseudo-effective R\mathbb{R}-Cartier divisor DD on XX, the relative non-nef locus equals the relative restricted base locus: NNef⁡(D/S)=B−(D/S)\operatorname{NNef}(D/S)=\mathbf{B}_{-}(D/S) as subsets of XX.

References

Primary source

arXiv

Additional references

algebraic-geometrybirational-geometrymixed-characteristic
Posted

Progress summary

Refreshed
Claimed solved

A new preprint claims a conditional proof in mixed characteristic, but the unrestricted conjecture remains unsettled and the proof has not been independently verified.

The conjecture concerns equality between relative non-nef loci and restricted base loci. No proposer or original date is identified in the available material.

September 2026 mixed-characteristic result

A preprint claims equality of the two loci under BCM-regularity and proves that BCM-regularity survives sufficiently small divisor perturbations. The result is a mixed-characteristic analogue of earlier characteristic-specific theorems, but its hypotheses restrict it to specified normal, flat, projective settings with singularity assumptions.

Current status (as of September 2026): A conditional mixed-characteristic theorem is claimed, while the unrestricted relative non-nef loci conjecture remains open and the claimed proof is unverified.

Sources

Solutions 0

No solutions have been posted yet.