Annales de l’institut Fourier (AIF)

Pila, Jonathan

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Jonathan PILA
A correction to “Counting rational points on a certain exponential-algebraic
surface”
Tome 67, no 3 (2017), p. 1277-1278.
<http://aif.cedram.org/item?id=AIF_2017__67_3_1277_0>
© Association des Annales de l’institut Fourier, 2017,
Certains droits réservés.
Cet article est mis à disposition selon les termes de la licence
CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 3.0 FRANCE.
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Article mis en ligne dans le cadre du
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Ann. Inst. Fourier, Grenoble
67, 3 (2017) 1277-1278
A CORRECTION TO “COUNTING RATIONAL POINTS
ON A CERTAIN EXPONENTIAL-ALGEBRAIC
SURFACE”
Annales de l’institut Fourier, vol. 60 (2010), n°2, 489–514
by Jonathan PILA (*)
Abstract. — We correct a mistake in the paper. All the theorems remain
correct.
Résumé. — Nous corrigeons une erreur dans la papier. Tous les théorèmes
restent vrais.
This note is to correct a part of the paper [1]. All the theorems remain
correct (though some of the implied constants will be adversely affected).
I am very grateful to John Armitage for pointing out my error.
The mistake occurs in the early lines of the proof of Theorem 6.1, p. 507.
The exhibited parameterization of the surface
X = {(x, y, z) ∈ (0, 1)3 : log x log y = − log z}
is not mild (as defined in the paper), as claimed there, because the function
z = exp
((
1− 1
sg
) (
1− 1
tg
))
, (s, t) ∈ (0, 1)2
is not mild. The seemingly innocuous change from −1/sg to 1− 1/sg (and
the same for t) to cover all of X spoils the mildness, and indeed the surface
needs to be non-trivially reparameterized near the x = 1 boundary for
small y (and near the y = 1 boundary for small x), as well as near the
origin.
Keywords: O-minimal structure, rational points.
Math. classification: 11G99, 03C64.
(*) Preparation of this correction was partially supported by a grant from EPSRC,
number EP/N008359/1.
1278 Jonathan PILA
We should therefore use instead the following parameterization of X by
four maps θi : (0, 1)2 → (0, 1)3, i = 1, . . . , 4 (here g ∈ Z, g > 0):
θ1(s, t) =
(
exp
(
− 1
sg
)
, exp
(
− 1
tg
)
, exp
(
− 1(st)g
))
,
θ2(s, t) =
(
exp (−sg), exp
(
− 1(st)g
)
, exp
(
− 1
tg
))
,
θ3(s, t) =
(
exp
(
− 1(st)g
)
, exp (−tg), exp
(
− 1
sg
))
,
θ4(s, t) = (exp (−sg) , exp (−tg) , exp (−(st)g)) .
These maps cover, respectively, the (x, y)-regions: (0, 1e )× (0,
1
e ), (
1
e , 1)×
(0, 1e ), (0,
1
e )×(
1
e , 1), and (
1
e , 1)×(
1
e , 1). The first three maps are (A, 1+1/g)-
mild for some suitable A = A(g) > 0, by Proposition 2.8, and the last is
(A(g), 0)-mild, being analytic on the closure of (0, 1)2. The omitted lines
are in the “algebraic part” of X and play no role.
BIBLIOGRAPHY
[1] J. Pila, “Counting rational points on a certain exponential-algebraic surface”, Ann.
Inst. Fourier 60 (2010), no. 2, p. 489-514.
Manuscrit reçu le 24 janvier 2017,
accepté le 1er mars 2017.
Jonathan PILA
Mathematical Institute,
University of Oxford,
Oxford, OX2 6RT (United Kingdom)
pila@maths.ox.ac.uk
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A correction to “Counting rational points on a certain exponential-algebraic surface”

Jiang, Wenshuai

Wang, Feng

Zhu, Xiaohua

Bergman Kernels for a sequence of almost Kähler–Ricci solitons

Ishii, Shihoko

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Shihoko ISHII
Corrigendum to “Mather discrepancy and the arc spaces”
Tome 67, no 4 (2017), p. 1609-1612.
<http://aif.cedram.org/item?id=AIF_2017__67_4_1609_0>
© Association des Annales de l’institut Fourier, 2017,
Certains droits réservés.
Cet article est mis à disposition selon les termes de la licence
CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 3.0 FRANCE.
http://creativecommons.org/licenses/by-nd/3.0/fr/
L’accès aux articles de la revue « Annales de l’institut Fourier »
(http://aif.cedram.org/), implique l’accord avec les conditions générales
d’utilisation (http://aif.cedram.org/legal/).
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Ann. Inst. Fourier, Grenoble
67, 4 (2017) 1609-1612
CORRIGENDUM TO “MATHER DISCREPANCY AND
THE ARC SPACES”
Annales de l’institut Fourier, vol. 63 (2013), n°1, 89–111
by Shihoko ISHII
Abstract. — This paper gives a correction of a theorem in “Mather discrep-
ancy and the arc spaces”.
Résumé. — Dans cette note nous corrigeons un théorème de « Discrépance de
Mather et les espaces d’arcs ».
In this paper, we make a correction of the statement of Theorem 4.7
in [2], where (v) was misstated as:
“The tangent cone of (X,x) has a reduced irreducible component.”
This statement should be corrected as:
“Let b : Y → X be the composite of the blow up b : Y → X at
the point x ∈ X and the normalization ν : Y → Y . Then, the fiber
scheme b−1(x) = E has a reduced irreducible component.”
The whole statement of the corrected theorem is as follows:
Theorem 4.7. — For a singularity (X,x) of dimension n the following
are equivalent:
(i) m̂ld(x;X,OX) = n;
(ii) λm = 0 for every m ∈ N;
(iii) λ0m = 0 for every m ∈ N;
(iv) λ01 = 0;
Keywords: singularities, arc space, log-canonical threshold, minimal log-discrepancy.
Math. classification: 14E18, 14B05.
1610 Shihoko ISHII
(v) Let b : Y → X be the composite of the blow up b : Y → X at the
point x ∈ X and the normalization ν : Y → Y . Then, the fiber
scheme b−1(x) = E has a reduced irreducible component, where
a reduced irreducible component means an irreducible component
which is reduced at the generic point.
Here, we give a proof for the relevant parts under this alteration.
Proof. — The proof in [2] of equivalence among (i), (ii) and (iii) is not
affected by the change in (v). The implication (iii) ⇒ (iv) is obvious. The
implication (iv)⇒ (v) is proved as follows:
Let E ⊂ Y be the scheme theoretic fiber of x by the blow up b : Y → X
Let g : Ỹ → Y be a log resolution of (Y,E) and let g be factored as
Ỹ
h−→ Y ν−→ Y .
Then, by the same argument in the corresponding part of the proof in the
paper [2], we obtain
dim g(E′reg) = n− 1 ,
where E′ is the scheme theoretic fiber of x by the morphism b ◦ g : Ỹ → X
and E′reg is the locus of non-singular points of E′. Therefore we obtain
dim h(E′reg) = n− 1 .
As h is isomorphic at the generic point of each irreducible component of
h(E′reg), this shows that E = b
−1(x) is reduced at an irreducible compo-
nent, which implies (v).
For the proof of (v)⇒(iii), we show that we can reduce the discussion
into the case that E has a reduced component and Y is non-singular at
the generic point of the component. Then the discussion in the proof of the
corresponding part in [2] would work.
Let E0 ⊂ Y be an irreducible component of E with the coefficient 1
in E and let E0 ⊂ Y be the irreducible components of E corresponding
to E0. Let e and e be the generic points of E0 and E0, respectively. The
normalization ν : Y → Y induces a homomorphism
ν̂∗ : ÔY,e → ÔY ,e .
of k-algebras. Let O0 be the image of ν̂∗:
ÔY,e  O0 ⊂ ÔY ,e .
Then SpecO0 is an analytic branch of Y at e that is dominated by
Spec ÔY ,e. Here, as Y is non-singular at e we have
ÔY ,e = K[[s]] ,
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM 1611
for some extension field K of k. For f =
∑
i ais
i ∈ K[[s]], we denote the
lowest degree i with ai 6= 0 by ords f and call it the order of f with respect
to the variable s.
We will show that O0 is a regular local ring. For that, we first prove that
O0 contains an element of order 1 with respect to s. Assume contrary, then
every element of O0 is either a unit or an element of order greater than 1.
Let ` ∈ ÔY,e be the defining equation of E in Y around e. We also denote
by ` the images of ` in O0 and in ÔY ,e by abuse of notation. Then, in
particular, ords ` > 2. As ` is also the defining equation of E in Y around
e by the assumption on E. Then, the above inequality shows that E is not
reduced at e, which yields a contradiction.
Now we may assume there is an element s′ ∈ O0 with order 1 with
respect to s. As ÔY ,e = K[[s]] = K[[s
′]], we may assume that s ∈ O0, by
replacing s by s′. By Cohen’s structure theorem, the residue field K ′ of the
complete local ring O0 is contained in O0 and therefore we obtain
K ′[[s]] ⊂ O0 .
Note that the base field k is of characteristic 0. Then the extensionK ′ ↪→ K
of fields is separable, therefore it is étale. Now as K ′[[s]] → K[[s]] is étale
and O0 → K[[s]] is flat, it follows that
K ′[[s]]→ O0
is étale by [1, IV, 17.7.7]. Therefore, O0 is also regular and ords(`) = 1.
Now one branch of Y at e is non-singular and E is reduced at the the
generic point. We restrict the discussion onto this branch. So, we may
assume that Y is non-singular at e and E is reduced at e. Then, the proof
of (v)⇒ (iii) in [2] completes the proof. 
Acknowledgments
Weichen Gu kindly provides us the following example which shows a
contradiction to the previous statement in the Theorem 4.7 in [2]. The
author would like to thank him.
Example. — Let X ⊂ A5 be a hypersurface defined by
y2 − x1x2x3x4 = 0 .
Then, the tangent cone has no reduced component, but (X, 0) satisfies (iv).
We should also note that X satisfies the condition (v).
TOME 67 (2017), FASCICULE 4
1612 Shihoko ISHII
BIBLIOGRAPHY
[1] A. Grothendieck, “Éléments de géométrie algébrique. IV: Étude locale des schémas
et des morphismes de schémas”, Publ. Math., Inst. Hautes Étud. Sci. 32 (1967), p. 1-
361.
[2] S. Ishii, “Mather discrepancy and the arc spaces”, Ann. Inst. Fourier 63 (2013),
no. 1, p. 89-111.
Manuscrit reçu le 10 août 2016,
accepté le 10 février 2017.
Shihoko ISHII
School of Art and Science,
Tokyo Woman’s Christian University,
Zenpukuji, Suginami, Tokyo (Japan)
shihoko@lab.twcu.ac.jp
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