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- Title
On a Conjecture of Bitoun and Schedler.
- Authors
Mustaţă, Mircea; Olano, Sebastián
- Abstract
Suppose that |$X$| is a smooth complex algebraic variety of dimension |$\geq 3$| and |$f$| defines a hypersurface |$Z$| in |$X$| , with a unique singular point |$P$|. Bitoun and Schedler conjectured that the |${\mathcal {D}}$| -module generated by |$\tfrac {1}{f}$| has length equal to |$g_{P}(Z)+2$| , where |$g_{P}(Z)$| is the reduced genus of |$Z$| at |$P$|. We prove that this length is always |$\geq g_{P}(Z)+2$| and equality holds if and only if |$\tfrac {1}{f}$| lies in the |${\mathcal {D}}$| -module generated by |$I_{0}(f)\tfrac {1}{f}$| , where |$I_{0}(f)$| is the multiplier ideal |${\mathcal {J}}(f^{1-\epsilon })$| , with |$0<\epsilon \ll 1$|. In particular, we see that the conjecture holds if the pair |$(X,Z)$| is log canonical. We can also recover, with an easy proof, the result of Bitoun and Schedler saying that the conjecture holds for weighted homogeneous isolated singularities. On the other hand, we give an example (a polynomial in |$3$| variables with an ordinary singular point of multiplicity |$4$|) for which the conjecture does not hold.
- Subjects
LOGICAL prediction
- Publication
IMRN: International Mathematics Research Notices, 2023, Vol 2023, Issue 21, p18254
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnac369