F-thresholds and Bernstein-Sato polynomials
Shunsuke Takagi (Kyushu University)
This is a joint work with M. Mustaţǎ and K.-i. Watanabe.
We introduce and study invariants of singularities in positive characteristic called
F-thresholds. They give an analogue of the jumping coefficients of multiplier ideals
in characteristic zero. Unlike these, however, the F-thresholds are not defined via
resolution of singularities, but via the action of the Frobenius map.
We will restrict ourselves to the case of an ambient nonsingular variety, when
our invariants have a down-to-earth description. Let (R, m) be a regular local ringof characteristic p > 0. We want to measure the singularities of a nonzero ideala ⊆ m. For every ideal J ⊆ m containing a in its radical, and for every e ≥ 1, wepute
νaJ (pe ) := max{r|ar 6⊆ J [p ] },eewhere J [p ] = (f p |f ∈ J) ⊂ R. One can check that the limit
νaJ (pe )e→∞pecJ (a) := limexists and is finite. We call this limit the F-threshold of a with respect to J. WhenJ = m, we simply write c(a) and νa (pe ). The invariant c(a) was introduced in [TW]under the name of F-pure threshold.
We point out that it is not known whether the analogue of two basic properties ofjumping coefficients of multiplier ideals hold in our setting: whether cJ (a) is always
a rational number and whether the set of all F-thresholds of a is discrete. Recently
Hara proved that this analogue holds true when R is a two-dimensional formal power
series ring over a finite field and a is a principle ideal (see [H]).
For simplicity, we assume that a and J are ideals in Z[X1 , . . . , Xn ], containedin (X1 , . . . , Xn ) and such that a is contained in the radical of J. Let us denote by
ap and Jp the localizations at (X1 , . . . , Xn ) of the images of a and J, respectively,in Fp [X1 , . . . , Xn ]. We want to compare our invariants mod p (which we write as
νaJ (pe ) and cJ (ap )) with the characteristic zero invariants of a (more precisely, withthe invariants around the origin of the image aQ of a in Q[X1 , . . . , Xn ]).
First, let us denote by lc0 (a) the log canonical threshold (which is the smallest
jumping coefficient in characteristic zero) of aQ around the origin. It follows fromresults of Hara and Watanabe (see [HW]) that if p À 0 then c(ap ) ≤ lc0 (a) andlimp→∞ c(ap ) = lc0 (a). Moreover, results of Hara and Yoshida from [HY] allow theextension of these formulas to higher jumping numbers. It is easy to give examples

in which c(ap ) 6= lc0 (a) for infinitely many p. On the other hand, one conjecturesthat there are infinitely many p with c(ap ) = lc0 (a).
A surprising fact is that our invariants for ap are related to the Bernstein-Satopolynomial ba,0 (s) of a. More precisely, we show that for all p À 0 and for all e,
we have ba,0 (νaJ (pe )) ≡ 0 (mod p). We show on some examples how to use this to
give roots of the Bernstein-Sato polynomial (and not just roots mod p). In theseexamples we will see the following behavior: given some ideal J containing a in its
radical, and e ≥ 1, we can find N such that for all i ∈ {1, . . . , N − 1} relatively
prime to N there are polynomials Pi ∈ Q[t] of degree e satisfying νaJ (pe ) = Pi (p) forall p À 0, with p ≡ i (mod N ). The previous observation implies that ba,0 (Pi (0)) is
divisible by p for every such p. By Dirichlet’s Theorem we deduce that Pi (0) is aroot of ba,0 .
An interesting question is which roots can be obtained by the above method.
It is shown in [BMS] that for monomial ideals the functions p → νaJ (pe ) behave as
described above, and moreover, all roots of the Bernstein-Sato polynomial are givenby this procedure. On the other hand, in general there are roots which cannot begiven by our method.

References[BMS] N. Budur, M. Mustaţǎ and M. Saito, Roots of Bernstein-Sato polynomials
for monomial ideals, preprint.
[H]

N. Hara, An approach to problems on F-jumping coefficients via p-fractals,
with an Appendix by P. Monsky, preprint.

[HW]N. Hara and K.-i. Watanabe, F-regular and F-pure rings vs. log terminal
and log canonical singularities, J. Algebraic Geom. 11 (2002), 363–392.

[HY]

N. Hara and K.-i. Yoshida, A generalization of tight closure and multiplierideals, Trans. Amer. Math. Soc. 355 (2003), 3143–3174.

[MTW] M. Mustaţǎ, S. Takagi and K.-i. Watanabe, F-thresholds and BernsteinSato polynomials, Proceedings of the 4th ECM, Stockolm, 2004, 341–364.
[TW]

S. Takagi and K.-i. Watanabe, On F-pure thresholds, J. Algebra 282 (2004),
278–297.