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Jumping coefficients of multiplier ideals. (English) Zbl 1061.14003
Let us consider a non singular complex variety \(X\) and some sheaf of ideals \(\mathbf a\). A log resolution of \(\mathbf a\) is a proper birational map \(\pi :X'\longrightarrow X\), such that \(X'\) is non singular, \({\mathbf a}{\mathcal O}_{X'}={\mathcal O}_{X'}(-F)\) is a normal crossing divisor and each component is smooth. For each real number \(c\) the multiplier ideal is defined by \({\mathcal I}(X,\mathbf a^c)=\pi_*({\mathcal O}_{X'}(K_{X'/X}-[cF]))\) where \(K_{X'/X}\) is the relative canonical divisor and [] means integer part. For \(x\in X\), it is clear that there exist some rational numbers \(\xi_i\) such that \({\mathcal I}(X,\mathbf a^c)_x={\mathcal I}(X,\mathbf a^{\xi_i})_x\) for all \(c\in [\xi_i,\xi_{i+1}[\). The numbers \(\xi_i\) are called jumping coefficients of multiplier ideals. In this interesting paper the authors prove that jumping coefficients appears naturally as deep invariants of singularities, for example:
Let \(f\) be a non zero polynomial and \(\xi\) a jumping coefficient of the ideal generated by \(f\), in the interval \((0,1]\), then \(-\xi\) is a zero of the Bernstein polynomial of \(f\).
The authors give a uniform Artin-Rees bound for multiplier ideals, specially the case of a principal ideal.
Reviewer’s remark: I should mention that the reviewer has constructed with many details a log resolution for integrally closed ideals and hypersurfaces non degenerated with respect to their Newton polyhedra, even for a complete intersection non degenerated with respect to their Newton polyhedra. Howald’s results follow from this easily; see M. Morales [Bull. Soc. Math. Fr. 112, 325–341 (1984; Zbl 0564.32006); Compos. Math. 64, 311–327 (1987; Zbl 0648.14005)].

MSC:
14B05 Singularities in algebraic geometry
32S05 Local complex singularities
13H05 Regular local rings
Biographic References:
Siu, Y.-T.
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