Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers

Kleiman, Steven L.; Ulrich, Bernd

doi:10.1090/s0002-9947-97-01960-0

Cited by 17 publications

(12 citation statements)

References 31 publications

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“…Symmetric resolutions in codimension 1. Resolutions of codimension 1 symmetric sheaves on P 3 have been studied fairly extensively by Barth [4], Casnati-Catanese [7], and Catanese [8] [9], in the context of surfaces with even sets of nodes and by Kleiman-Ulrich [23] in the context of self-linked curves. The next theorem, conjectured by Barth and Catanese, was proven by Casnati-Catanese for symmetric sheaves on P 3 ([7] Theorem 0.3).…”

Section: Codimension One Sheavesmentioning

confidence: 99%

Lagrangian subbundles and codimension 3 subcanonical subschemes

Eisenbud¹,

Popescu²,

Walter³

2001

Duke Math. J.

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We show that a Gorenstein subcanonical codimension 3 subscheme Z ⊂ X = P N , N ≥ 4, can be realized as the locus along which two Lagrangian subbundles of a twisted orthogonal bundle meet degenerately, and conversely. We extend this result to singular Z and all quasiprojective ambient schemes X under the necessary hypothesis that Z is strongly subcanonical in a sense defined below. A central point is that a pair of Lagrangian subbundles can be transformed locally into an alternating map. In the local case our structure theorem reduces to that of Buchsbaum-Eisenbud [6] and says that Z is Pfaffian.We also prove codimension one symmetric and skew-symmetric analogues of our structure theorems.

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Section: Codimension One Sheavesmentioning

confidence: 99%

Lagrangian subbundles and codimension 3 subcanonical subschemes

Eisenbud¹,

Popescu²,

Walter³

2001

Duke Math. J.

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“…B (J −1 ) = 1, so J −1 is a grade one perfect B-module. By [KU,Proposition 3.6], J is a grade one perfect B-module, soJ is a grade two perfect ideal. On the other hand, depth B (B/J) = 1 + depth B (B/(Q +P )).…”

Section: Katzmentioning

confidence: 99%

On the existence of maximal Cohen-Macaulay modules over 𝑝th root extensions

Katz

1999

Proc. Amer. Math. Soc.

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“…Mond and Pellikaan [29, 4.3 p. 131] went on to prove that, if N 3 has codimension in Y at least 3, then N 3 has codimension exactly 3 and N 3 is Cohen-Macaulay because then Fitt Y 2 (f * O X ) y is locally a symmetric determinantal ideal. Those results will not be recovered in this paper; however, see [22] where there are new proofs, which, unlike the old, work in the present general setting, and there are related new proofs of the characterization (due to Valla and Ferrand) of perfect self-linked ideals of grade 2 in an arbitrary Noetherian local ring as the ideals of maximal minors of suitable n by n − 1 matrices having symmetric n − 1 by n − 1 subblocks.…”

Section: Introductionmentioning

confidence: 91%

The multiple-point schemes of a finite curvilinear map of codimension one

Kleiman¹,

Lipman

Ulrich

1996

Ark. Mat.

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Let X and Y be smooth varieties of dimensions n−1 and n over an arbitrary algebraically closed field, f : X → Y a finite map that is birational onto its image. Suppose that f is curvilinear; that is, for all x ∈ X, the Jacobian ∂f (x) has rank at least n−2. For r ≥ 1, consider the subscheme N r of Y defined by the (r − 1)-th Fitting ideal of the O Y -module f * O X , and set M r := f −1 N r . In this setting-in fact, in a more general setting-we prove the following statements, which show that M r and N r behave like reasonable schemes of source and target r-fold points of f .If each component of M r , or equivalently of N r , has the minimal possible dimension n − r, then M r and N r are Cohen-Macaulay, and their fundamental cycles satisfy the relation, f * [M r ] = r[N r ]. Now, suppose that each component of M s , or of N s , has dimension n − s for s = 1, . . . , r + 1. Then the blowup Bl(N r , N r+1 ) is equal to the Hilbert scheme Hilb r f , and the blowup Bl(M r , M r+1 ) is equal to the universal subscheme Univ r f of Hilb r f × Y X; moreover, Hilb r f and Univ r f are Gorenstein. In addition, the structure map h: Hilb r f → Y is finite and birational onto its image; and its conductor is equal to the ideal J r of N r+1 in N r , and is locally self-linked. Reciprocally, h * O Hilb r f is equal to Hom(J r , O N r ). Moreover, h * [h −1 N r+1 ] = (r + 1)[N r+1 ]. Similar assertions hold for the structure map h 1 : Univ r f → X if r ≥ 2.1980 Mathematics Subject Classification (1985 Revision). Primary 14C25; Secondary 14O20, 14N10.Acknowledgements. It is a pleasure to thank Luchezar Avramov, Winfried Bruns, David Eisenbud, Hans-Bjørn Foxby, and Christian Peskine for fruitful discussions. Avramov and Foxby discussed Gorenstein maps; Bruns and Eisenbud discussed properties of Fitting ideals; and Peskine explained at length his work [12] with Gruson.

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Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers

Cited by 17 publications

References 31 publications

Lagrangian subbundles and codimension 3 subcanonical subschemes

Lagrangian subbundles and codimension 3 subcanonical subschemes

On the existence of maximal Cohen-Macaulay modules over 𝑝th root extensions

The multiple-point schemes of a finite curvilinear map of codimension one

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