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

Let PXP\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 rjIr_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

mldP(X,a)=mldP(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.

Sources & referencesView supporting material

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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