# An invariant detecting rational singularities via the log canonical threshold

@article{Cluckers2019AnID, title={An invariant detecting rational singularities via the log canonical threshold}, author={Raf Cluckers and Mircea Mustaţǎ}, journal={arXiv: Algebraic Geometry}, year={2019} }

We show that if f is a nonzero, noninvertible function on a smooth complex variety X and J_f is the Jacobian ideal of f, then lct(f, J_f^2)>1 if and only if the hypersurface defined by f has rational singularities. Moreover, if this is not the case, then lct(f, J_f^2)=lct(f). We give two proofs, one relying on arc spaces and one that shows that the minimal exponent of f is at least as large as lct(f, J_f^2). In the case of a polynomial over the algebraic closure of Q, we also prove an analogue…

## 2 Citations

IGUSA’S CONJECTURE FOR EXPONENTIAL SUMS: OPTIMAL ESTIMATES FOR NONRATIONAL SINGULARITIES

- MathematicsForum of Mathematics, Pi
- 2019

We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa’s conjecture on exponential…

Hodge ideals and minimal exponents of ideals

- Mathematics
- 2019

We define and study Hodge ideals associated to a coherent ideal sheaf J on a smooth complex variety, via algebraic constructions based on the already existing concept of Hodge ideals associated to…

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IGUSA’S CONJECTURE FOR EXPONENTIAL SUMS: OPTIMAL ESTIMATES FOR NONRATIONAL SINGULARITIES

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We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa’s conjecture on exponential…

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