Mustaţă's uniform m-adic semicontinuity conjecture for minimal log discrepancies

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Let P∈XP\in X be the germ of a smooth threefold, let m\mathfrak{m} be the maximal ideal in OX\mathscr{O}_X defining PP, and let

a=∏j=1eajrj,b=∏j=1ebjrj\mathfrak{a}=\prod_{j=1}^e\mathfrak{a}_j^{r_j},\qquad \mathfrak{b}=\prod_{j=1}^e\mathfrak{b}_j^{r_j}

be R\mathbf{R}-ideals on XX. Fix a finite subset II of the positive real numbers. Mustaţă's uniform m\mathfrak{m}-adic semicontinuity conjecture. There exists a positive integer ll depending only on II such that, whenever rj∈Ir_j\in I and aj+ml=bj+ml\mathfrak{a}_j+\mathfrak{m}^l=\mathfrak{b}_j+\mathfrak{m}^l for every jj, one has

mld⁡P(X,a)=mld⁡P(X,b).\operatorname{mld}_P(X,\mathfrak{a})=\operatorname{mld}_P(X,\mathfrak{b}).

The conjecture asserts uniform determination of the minimal log discrepancy by sufficiently high-order truncations of the ideals; the source attributes it originally to Mustaţă and lists it among conjectures equivalent to the ACC conjecture.

References

Primary source

Masayuki Kawakita, “On equivalent conjectures for minimal log discrepancies on smooth threefolds”, arXiv:1803.02539 (2018).

Additional references

4 papers in this index state this conjecture (2010–2018). The statement above is taken from the most recent of them; the others are arXiv:1305.1410, arXiv:1205.6014, arXiv:1012.0395.

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