Correction to: Conormal Spaces and Whitney Stratifications

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-022-09602-7 CORRECTION Correction to: Conormal Spaces and Whitney Stratifications Martin Helmer1 · Vidit Nanda2 © SFoCM 2022 Abstract This note remedies an error in our paper tilted Conormal Spaces and Whitney Stratifications (Found. Comput. Math., 2022). Correction to: Found Comput Math https://doi.org/10.1007/s10208-022-09574-8 Introduction We are grateful to the research group comprising Lihong Zhi, Nan Li, Zhihong Yang and Zijia Li for alerting us to an error in [5]: they discovered that the Macaulay2 software package accompanying that paper did not output the origin as a separate stratum when called with the projective Whitney cusp as input. Our goal in this note is to explain and fix the error, which originated from a misinterpretation of the algebraic criterion [3, Remark 4.11] that underlies the algorithms of [5]. The statement of the criterion—taken from [3, Remark 4.11] and reproduced below—paraphrases [7, Proposition 1.3.8]. Lemma 0.1 Let Z be an analytic variety with conormal map κ Z : Con(Z )  Z , and let S ⊂ Z be a smooth analytic subset. The pair (Z reg , S) satisfies Whitney’s Condition (B) if and only if we have the containment I [Con(Z ) ∩ Con(S)] ⊂ I [κ Z−1 (S)] (1) The original article can be found online at https://doi.org/10.1007/s10208-022-09574-8. B Vidit Nanda Martin Helmer 1 Department of Mathematics, North Carolina State University, Raleigh, NC, USA 2 Mathematical Institute, University of Oxford, Oxford, UK 123 Foundations of Computational Mathematics of ideal sheaves, where the right side is an integral closure. Fix a projective variety X ⊆ Pn and an irreducible subvariety Y of its singular locus X sing . The proof of [5, Theorem 4.3] uses Lemma 0.1 in the case where Z = X and S = Y − A, with A being a certain proper subvariety of Y that contains Ysing . It turns out, however, that the Lemma only holds in this case if S = Y − A is closed (namely, if A is empty). To produce the correct variant of (1) when S is open, we are compelled to return to the original statement of [7, Proposition 1.3.8]. Outline In Sect. 1, we derive a version of Lemma 0.1 for the case where S is open directly from [7] in Sect. 1. We rectify our proof of [5, Theorem 4.3] in Sect. 2, and describe the concomitant modifications to [5] in Sect. 3. These corrections have been fully incorporated in the arXiv version of our paper [6]. 1 The Correct Interpretation We work throughout in a polynomial ring R which will be either be the coordinate ring of Pn or of Pn × (Pn )∗ . 1.1 Notation, etc. The following conventions are adopted in the sequel: (1) When considering a closed subscheme Z of either ambient space, we will denote its defining ideal sheaf by I [Z ] — although this is simply an R-ideal, the notation serves to emphasize that this ideal may (and often will) fail to be radical. (2) In contrast, the radical ideal associated to a variety V in either of our ambient spaces will be written I V . We also write Vsing for the singular locus, and Vreg = V − Vsing for the regular points. (3) We write I [Z ] to indicate the integral closure of I [Z ]. (4) Given a subvariety W ⊂ V and the conormal map κV : Con(V )  V of V , the ideal sheaf of the closed scheme κV−1 (V − W ) is denoted I [κV−1 (V − W )]; this equals the intersection of all primary components of I [κV−1 (V )] that are not supported on κV−1 (W ). (5) Finally, we recall that for a polynomial ideal I  R with primary decomposition I = Q 1 ∩ · · · ∩ Q r , the set Assoc(I ) of I ’s associated primes√consists of√prime ideals given by taking the radical of each primary component { Q 1 , . . . , Q r }. 1.2 An Algorithmic Criterion for Condition (B) The following result is [7, Prop 1.3.8], specialized to our situation and using the above notation.1 1 In [7] the authors consider conormal spaces relative to an analytic map f : X → S between smooth analytic spaces, and obtain a regularity criterion denoted Condition (w f ). This criterion reduces to the standard Condition (B) when S is a point [7, Remark 1.3.9]. We have therefore discarded the relative perspective in our rephrasing of [7, Proposition 1.3.8]. 123 Foundations of Computational Mathematics Lemma 1.1 Let X ⊂ Pn be a projective variety with conormal map κ X : Con(X )  X , and consider a subvariety Y of X satisfying Y ⊂ X sing . Let I p be the maximal ideal of a point p ∈ Yreg considered in the coordinate ring of Pn × (Pn )∗ . Then Condition (B) holds for the pair (X reg , Yreg ) at p if and only if we have a containment     Assoc I [Con(X ) ∩ Con(Y )] + I p ⊃ Assoc I [κ X−1 (Y )] + I p of associated primes. If A ⊂ Y is a closed proper subvariety that contains Ysing but not the point p, then the above criterion for (X reg , Yreg ) gives an identical criterion for the pair (X reg , Y − A) at p. Our goal here is to derive from this local result a global algorithmic criterion for checking whether or not (X reg , Y − A) satisfies Condition (B) at all points y ∈ Y − A. In the statement below, I [κ X−1 (Y − A)], respectively I [κ X−1 (Y − A)], denotes the intersection of all primary components of I [κ X−1 (Y )], respectively I [κ X−1 (Y )], which are not supported on κ X−1 (A). Lemma 1.2 Let X , Y be as defined in the statement of Lemma 1.1 and let A  Y be a subvariety with Ysing ⊂ A. Then Condition (B) holds for all points in the pair (X reg , Y − A) if and only if we have the containment     Assoc I [Con(X ) ∩ Con(Y )] + I [κ X−1 (Y − A)] ⊃ Assoc I [κ X−1 (Y − A)] (2) of associated primes. Proof Set Y = Y − A. We know from Lemma 1.1 that Condition (B) holds for (X reg , Y ) at a point p ∈ Y if and only if we have the containment     Assoc I [Con(X ) ∩ Con(Y )] + I p ⊃ Assoc I [κ X−1 (Y )] + I p . (3) We now claim that requiring such a containment for each p in Y , is equivalent to the containment in (2). Since p is not in A by assumption, we have that κ A−1 ( p) is not contained in any primary components supported on κ −1 (A) and hence, when summing with I p , we can replace I [κ X−1 (Y − A)] on the left side of (2) by I [κ X−1 (Y )]. Similarly, when summing with I p , the integral closure on the right side may as well be replaced by I [κ X−1 (Y )]. Recall that I [κ X−1 (Y − A)] is the ideal sheaf of the scheme κ X−1 (Y − A); thus, if we have containment (3) for every p ∈ Y , then we automatically have the corresponding containment for the Zariski closures, hence we obtain (2). Conversely, suppose that (2) holds and consider any p ∈ / A. All associated primes of I [κ X−1 (Y − A)]+I p arise from summing I p with associated primes of I [κ X−1 (Y − A)]. Since all these primes are also contained in the left hand set, and since we have 123 Foundations of Computational Mathematics I [κ X−1 (Y − A)] + I p = I [κ X−1 (Y )] + I p = ICon(X ) + I p we obtain the desired containment (3). 2 Rectifying the Argument Let R be the coordinate ring of Pn × (Pn )∗ , and consi (...truncated)