26 research outputs found
An estimate for F-jumping numbers via the roots of the Bernstein-Sato polynomial
Given a smooth complex algebraic variety and a nonzero regular function
on , we give an effective estimate for the difference between the
jumping numbers of and the -jumping numbers of a reduction of
to characteristic , in terms of the roots of the Bernstein-Sato
polynomial of . As an application, we show that if has no roots
of the form , with a positive integer, then the -pure
threshold of is equal to the log canonical threshold of for
with .Comment: 10 pages; v.2: using a bound for the roots of the Bernstein-Sato
polynomial, we deduce a uniform estimate only involving the dimension of the
ambient variety and explain how this extends to possibly non-principal ideal
Graded Betti numbers of general finite subsets of points on projective varieties
See directly the article
The log canonical threshold and rational singularities
We show that if is a nonzero, noninvertible function on a smooth complex
variety and is the Jacobian ideal of , then
if and only if the hypersurface defined by has rational singularities.
Moreover, if it does not have rational singularities, then . We give two proofs, one relying on arc spaces and
one that goes through the inequality , where is the minimal exponent of .
In the case of a polynomial over , we also prove an
analogue of this latter inequality, with replaced by
the motivic oscillation index . We also show a part of Igusa's
strong monodromy conjecture, for poles larger than . We
end with a discussion of lct-maximal ideals: these are ideals with the
property that for every with .Comment: This supersedes arXiv:1901.08111. V.2: final version, to appear in
the special volume of Alg. Geom. Phys. in honor of Yuri I. Mani
Valuations and asymptotic invariants for sequences of ideals
We study asymptotic jumping numbers for graded sequences of ideals, and show
that every such invariant is computed by a suitable real valuation of the
function field. We conjecture that every valuation that computes an asymptotic
jumping number is necessarily quasi-monomial. This conjecture holds in
dimension two. In general, we reduce it to the case of affine space and to
graded sequences of valuation ideals. Along the way, we study the structure of
a suitable valuation space.Comment: 49 pages; v2: we now work more generally in the setting of arbitrary
excellent regular schemes; some details regarding the definition of
quasi-monomial valuations have been added in Section 3.1; v3: minor changes,
this is the final version, to appear in Ann. Inst. Fourier (Grenoble
A Frobenius variant of Seshadri constants
We define and study a version of Seshadri constant for ample line bundles in
positive characteristic. We prove that lower bounds for this constant imply the
global generation or very ampleness of the corresponding adjoint line bundle.
As a consequence, we deduce that the criterion for global generation and very
ampleness of adjoint line bundles in terms of usual Seshadri constants holds
also in positive characteristic.Comment: 16 page