• Corpus ID: 119720876

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

References

SHOWING 1-10 OF 20 REFERENCES
Onb-function, spectrum and rational singularity
by duality [18], because R~Tr,CCy, = 0 for i > 0 by [6, 31] (this follows also from [13, 21, 23]) where rc is assumed projective. Here COy denotes the dualizing sheaf (i.e., the dualizing complex
ZETA FUNCTIONS FOR ANALYTIC MAPPINGS, LOG-PRINCIPALIZATION OF IDEALS, AND NEWTON POLYHEDRA
In this paper we provide a geometric description of the possi- ble poles of the Igusa local zeta function Z�(s, f) associated to an analytic mapping f = (f1, . . . , fl) : U(� K n ) ! K l , and a
Germs of arcs on singular algebraic varieties and motivic integration
We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the
Hodge ideals for Q-divisors, V-filtration, and minimal exponent
We explicitly compute the Hodge ideals of Q-divisors in terms of the V-filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties
IGUSA’S CONJECTURE FOR EXPONENTIAL SUMS: OPTIMAL ESTIMATES FOR NONRATIONAL SINGULARITIES
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
Contact loci in arc spaces
We give a geometric description of the loci in the arc space defined by order of contact with a given subscheme, using the resolution of singularities. This induces an identification of the
Arcs on determinantal varieties
We study arc spaces and jet schemes of generic determinantal varieties. Using the natural group action, we decompose the arc spaces into orbits, and analyze their structure. This allows us to compute
The fundamental theorem of prehomogeneous vector spaces modulo $p^m$ (With an appendix by F. Sato)
(Theoreme fondamental des espaces vectoriels prehomogenes modula p m . Avec un appendice par F. Sato) Soit K un corps de nombres avec anneaux d'entiers O K; nous prouvons un analogue, sur des anneaux
ASYMPTOTIC HODGE STRUCTURE IN THE VANISHING COHOMOLOGY
The asymptotics of integrals which depend on a critical point of a holomorphic function and the mixed Hodge structure in the vanishing cohomology are compared. Bibliography: 38 titles.
IMPANGA lecture notes on log canonical thresholds
These are lecture notes from the IMPANGA 2010 Summer school. They give an introduction to log canonical thresholds, covering some basic properties, examples, and some recent results and open
...
...