Urbano's conjecture on uniformly generated divisorial sheaves

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Let XX be a projective normal variety, and let D∈WDivQ(X)D\in {\rm WDiv}_{\mathbb{Q}}(X) be a divisordivisor. A divisor AA on XX is very ample if its complete linear system defines a closed embedding of XX into projective space.

Urbano's conjecture. There exists a very ample divisor AA such that OX(mD)⊗OX(A)⊗m\mathcal{O}_X(mD)\otimes \mathcal{O}_X(A)^{\otimes m} is globally generated for every m≥1m\geq 1.

The conjecture asserts a uniform linear ample twist that makes all positive multiples of a rational Weil divisor globally generated. The supplied text recalls it from Urbano; no resolution is given here.

References

Primary source

Stefano Urbinati, “Divisorial models of normal varieties”, arXiv:1211.1692 (2015).

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