Fitting ideals and multiple points of analytic mappings

Mond, David; Pellikaan, G.R.

doi:10.1007/bfb0090254

Cited by 66 publications

(67 citation statements)

References 49 publications

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“…The correspondence is summarized formally in Theorem 3.7. These results extend the corresponding work of Mond and Pellikaan [26] and of de Jong and van Straten [7], with the notable exception of the characterization of the Gorenstein B, which is new.…”

Section: Introductionsupporting

confidence: 88%

“…Indeed, Proposition 3.8 asserts that, if B is Gorenstein, then ∆ defines an isomorphism, R/F 2 ∼ −→ I (2) /I 2 . This result was proved under the additional hypothesis that F 1 is radical by Mond and Pellikaan [26,Thm. 4.4,p.…”

Section: Introductionmentioning

confidence: 71%

“…Part (2) asserts that the height of F 2 is at most 4 if B is CohenMacaulay and is of rank 2 over R/(∆), where ∆ is a nonzerodivisor, and if R is regular. Part (1) for i = 2 was proved (implicitly) by Mond and Pellikaan [26,Thm. 4.3,p.…”

Section: Introductionmentioning

confidence: 89%

“…In Part (1) the matrix is symmetric, and in Part (2) it's alternating. In their case, Mond and Pellikaan [26,Prop. 4.1,p.…”

Section: Introductionmentioning

confidence: 98%

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

Kleiman

Ulrich

1997

Trans. Amer. Math. Soc.

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Abstract. Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade 2 as those with a Hilbert-Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade 1 can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade 1 that are birational onto their image, on the one hand, and self-linked perfect ideals of grade 2 that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.

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Section: Introductionsupporting

confidence: 88%

Section: Introductionmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 89%

“…In Part (1) the matrix is symmetric, and in Part (2) it's alternating. In their case, Mond and Pellikaan [26,Prop. 4.1,p.…”

Section: Introductionmentioning

confidence: 98%

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

Kleiman

Ulrich

1997

Trans. Amer. Math. Soc.

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“…It is known (see, for example [15,Theorem 3.4]) that F 1 is a determinantal ideal of height 2, and it follows from the long exact sequence of Ext arising from the short exact sequence (2) that the inclusion…”

Section: Proofmentioning

confidence: 99%

Stable mappings and logarithmic relative symplectic forms

Holland

Mond

1999

Math Z

Self Cite

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We use Lempert's version of Riemann mapping to construct non-equivalent symplectic forms on an ellipsoid in C n. Any bounded pseudoconvex domain D ⊆ C n admits a strictly plurisubharmonic exhausting function u. If u is smooth (or at least C 2) then ω = dd c u = 2i∂ ¯ ∂u is a symplectic form on D. The following theorem was proved in [2]. Theorem A (Eliashberg, Gromov). Let u 1 , u 2 be two smooth strictly plurisubhar-monic exhausting functions on a bounded strictly pseudoconvex domain D: D = {u 1 < ∞} = {u 2 < ∞}. Then the forms ω 1 = dd c u 1 and ω 2 = dd c u 2 are symplectomorphic, i.e. there exists a diffeomorphism ψ : D → D such that ψ * ω 2 = ω 1. As u i (z) → ∞ for z → ∂D it can easily be seen that the volume of D with respect to the volume forms Ω i = ω i ∧ · · · ∧ ω i (i = 1, 2) is infinite. We will construct an example which shows that Theorem A no longer holds true for the finite volume case, i.e. when we give the domain not by exhausting but by defining functions. Let D be a bounded strictly pseudoconvex domain represented by a strictly plurisubharmonic smooth defining function u 1 , i.e. D = {u 1 (z) < 0} ⊆ C n with du 1 | ∂D = 0. D is a symplectic manifold with the 2-form ω 1 = dd c u 1. Observe that the volume of D with respect to the volume form Ω 1 = ω 1 ∧ · · · ∧ ω 1 is finite. If u 2 is another defining function for D that is equal to u 1 in a neighbourhood of the boundary ∂D then D Ω 1 = D Ω 2 as follows from Stokes' theorem. Moreover, using Moser's deformation argument (see e.g. [1], p.20) we can easily construct a symplectomorphism between (D, ω 1) and (D, ω 2). If u 1 and u 2 are not equal in a neighbourhood of ∂D the volume condition alone is not sufficient to guarantee the existence of a symplectomorphism as will be shown in the following example. The basis of our construction is Lempert's theorem stating that for any strictly convex domain D ⊆ C n there is a homeomorphism of the unit ball ρ : B → D which has many similar properties to the usual Riemann mapping in C. The property we need here is that the function u : D → R, u(z) = 1 4 (|ρ −1 (z)| 2 − 1) is a plurisubharmonic defining function of D and moreover the relation dd c u = ρ −1 * ω 1 holds where ω 1 = dy i ∧ dx i = 1 4 dd c (|z| 2 − 1) is the standard symplectic form in C n (see e.g. [5], Lemma 2.6).

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Mond's conjecture for maps between curves

Ament

Nuño–Ballesteros

2017

Mathematische Nachrichten

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A theorem by D. Mond shows that if ∶ (ℂ, 0) → ( ℂ 2 , 0 ) is finite and has has degree one onto its image ( , 0), then the A -codimension is less than or equal to the image Milnor number ( ), with equality if and only if ( , 0) is weighted homogeneous.Here we generalize this result to the case of a map germ ∶ ( , 0), where ( , 0) is a plane curve singularity. K E Y W O R D SA -codimension, image Milnor number, curve singularities M S C ( 2 0 1 0 ) Primary: 32S30; Secondary: 32S05, 58K60

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Fitting ideals and multiple points of analytic mappings

Cited by 66 publications

References 49 publications

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

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

Stable mappings and logarithmic relative symplectic forms

Mond's conjecture for maps between curves

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