Erratum: article Melles-Milman, Lemma 4.5, p. 719-720, fascicule 4, 2006

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

CAROLINEGRANTMELLES, PIERREMILMAN

Erratum: article Melles-Milman, Lemma 4.5, p. 719-720, fascicule 4, 2006

Tome XVI, n^o2 (2007), p. 425-426.

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ERRATUM

(article Melles-Milman, Lemma 4.5, p. 719-720, fascicule 4, 2006)

”Due to an error of the authors, a few lines were missing in the fourth paragraph of the original proof of Lemma IV.5 of our recently published paper, leaving a gap in the proof.”

Lemma 4.5. — The same polynomials that generate J(0,0) also gener- ate J over a neighborhood of (0,0) in U ×Cⁿ⁺¹ and generate I over a neighborhood of {0} ×Pⁿ inU×Pⁿ.

Proof. — Suppose that J is generated in a neighborhood of (0,0) by F₁(x, y), ..., F_s(x, y), whereF_i(x, y) is a homogeneous polynomial of degree di in y with analytic coefficients inx. We will show thatI is generated on a neighborhood of {0} ×Pⁿ in U×Pⁿ by the corresponding polynomials Fi(x, ξ), where [ξ] = [ξ0:...:ξn] are homogeneous coordinates forPⁿ. More precisely, we will show thatI is generated on a neighborhood of any point q∈ {0} ×Pⁿ by dehomogenizations ofF1, ..., Fsnear q.

Choose homogeneous coordinates ξ on Pⁿ such that q = (0,[1 : 0 : ... : 0]). Nonhomogeneous coordinates on the set W ={ξ₀ = 0} ⊂Pⁿ are w_i= ^ξ_ξⁱ

0 for 1in. We will check thatI is generated in a neighborhood ofqby the polynomials

F_i(x, ξ) ξ₀^di =F_i

x, ξ

ξ0

=F_i(x,1, w₁, ..., w_n).

First we look at the mapsσ₁ andσ₂ in local coordinates. We may use (x, y₀, w) as local coordinates inσ⁻₂¹(U×W)∼=U×C×W. Local coordinates forU×Cⁿ⁺¹are (x, y₀, y₁, ..., y_n), wherey_i=y₀w_i for 1in. The maps σ₁andσ₂ are given by

σ₁(x, y₀, w) = (x, y₀, y₀w) and σ₂(x, y₀, w) = (x, w).

Suppose that G is a holomorphic section of I on a neighborhood of q in U ×Pⁿ. Then σ₂^∗G is a holomorphic section of σ⁻¹₂ I in a neighbor- hood of σ₂⁻¹(q) = {(0, y0,0) : y0 ∈ C}. The homogeneous polynomials F1, ..., Fsthat generate J(0,0)also generate J =σ1∗(σ₂⁻¹I) on a neighbor- hood of (0,0)∈U ×Cⁿ⁺¹ (since J is coherent, by the Direct Image The- orem), so their pullbacks σ₁^∗F1, ..., σ₁^∗Fs generate ˜J := σ₁⁻¹J on a neigh- borhood of σ⁻₁¹(0,0) ∈ U ×C˜ⁿ⁺¹ and therefore generate ˜I := σ₂⁻¹I off

– 425 –

Erratum

H :=σ₁⁻¹(U× {0}). Hence Supp(˜I/J˜)⊂H and therefore (by the complex analytic nullstellensatz) there exist an integerd >0 and holomorphic func- tions A₁, ..., A_s on a neighborhood of the point (x= 0, y₀ = 0, w = 0) in U×C˜ⁿ⁺¹ such that

y₀^dσ₂^∗G(x, y₀, w) =

s

i=1

A_i(x, y₀, w)σ₁^∗F_i(x, y₀, w)

on that neighborhood. But σ^∗₂G(x, y₀, w) =G(x, w) is independent of the value ofy₀ and σ^∗₁F_i(x, y₀, w) =F_i(x, y₀, y₀w) =y₀^diF_i(x,1, w) since F_i is homogeneous of degreed_i iny. Therefore, by comparing terms iny^d₀ of both sides of the equation above, it follows that there are holomorphic functions a1, ..., as (on a neighborhood of (x= 0, y0= 0, w = 0)) depending only on xandwsuch that

G(x, w) =

s

i=1

ai(x, w)Fi(x,1, w).

Since the functions Fi(x,1, w) are the local dehomogenizations of the ho- mogeneous polynomialsF(x, ξ), we are done.

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