The Jacobian algebra of a graded Gorenstein singularity

Wahl, Jonathan

doi:10.1215/s0012-7094-87-05540-2

Cited by 88 publications

(57 citation statements)

References 22 publications

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“…J. Wahl pointed out that the "if" part of Conjecture 2.4 holds in the complex-analytic category. This follows from a result of Naruki [Nar,2.1] (see [Wahl,(1.2)]) and Proposition 2.2.…”

Section: Corollary 23 Suppose R Is a Nonregular Integrally Closed Gmentioning

confidence: 68%

Almost split sequences and Zariski differentials

Martsinkovsky¹

1990

Trans. Amer. Math. Soc.

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ABSTRACT. Let R be a complete two-dimensional integrally closed analytic kalgebra. Associated with R is the Auslander module A from the fundamental sequence 0 -+ W R -+ A -+ R -+ k -+ 0 and the module of Zariski differentials Dk(R)** . We conjecture that these modules are isomorphic if and only if R is graded. We prove this conjecture for (a) hypersurfaces f = X; + g(X1 ' X 2 ) , (b) quotient singularities, and (c) R graded Gorenstein.

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“…J. Wahl pointed out that the "if" part of Conjecture 2.4 holds in the complex-analytic category. This follows from a result of Naruki [Nar,2.1] (see [Wahl,(1.2)]) and Proposition 2.2.…”

Section: Corollary 23 Suppose R Is a Nonregular Integrally Closed Gmentioning

confidence: 68%

Almost split sequences and Zariski differentials

Martsinkovsky¹

1990

Trans. Amer. Math. Soc.

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“…We will also suppose everywhere that the projective embedding of X is subcanonical, so that ω X ∼ = O X (m) for some integer m. Key references for most of the considerations below are [31], [36]. We stress that their results hold in the more general case of an affine isolated singularity, but here we only consider the case of affine cones over smooth projective varieties.…”

Section: The Infinitesimal Deformation Module Of An Affine Isolated Smentioning

confidence: 99%

Hodge theory and deformations of affine cones of subcanonical projective varieties

Natale¹,

Fatighenti

Fiorenza

2017

J. London Math. Soc.

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Abstract. We investigate the relation between the Hodge theory of a smooth subcanonical n-dimensional projective variety X and the deformation theory of the affine cone AX over X. We start by identifying H n−1,1 prim (X) as a distinguished graded component of the module of first order deformations of AX, and later on we show how to identify the whole primitive cohomology of X as a distinguished graded component of the Hochschild cohomology module of the punctured affine cone over X. In the particular case of a projective smooth hypersurface X we recover Griffiths' isomorphism between the primitive cohomology of X and certain distinguished graded components of the Milnor algebra of a polynomial defining X. The main result of the article can be effectively exploited to compute Hodge numbers of smooth subcanonical projective varieties. We provide a few example computation, as well a SINGULAR code, for Fano and Calabi-Yau threefolds.

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“…The next thing to determine is T 2 , the space where the obstructions lie. For T 2 (ν), ν < −2, we have a general vanishing result [Wahl 1987], and it is not difficult to find the dimension in case ν > −2. We find the dimension of T 2 with the Main Lemma of [Behnke-Christophersen 1991], which connects the number of generators of T 2 with the codimension of smoothing components in the base space of the versal deformation of a general hyperplane section.…”

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confidence: 91%

Deformations of cones over hyperelliptic curves.

1996

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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The deformation theory of a two-dimensional singularity, which is isomorphic to an affine cone over a curve, is intimately linked with the (extrinsic) geometry of this curve. In recent times various authors have studied one-parameter deformations, partly under the guise of extensions of curves to surfaces (cf. the survey [Wahl 1989]). In this paper we consider the versal deformation of cones, in the simplest case: cones over hyperelliptic curves of high degree. In particular, we show that for degree 4g + 4, the highest degree for which interesting deformations exist, the number of smoothing components is 2 2g+1 (the case g = 3 is exceptional). Let X be the cone over a hyperelliptic curve C, embedded with a line bundle . This implies that all deformations in negative degree must be obstructed. If S is a surface with C as hyperplane section, then one can degenerate S to the projective cone over C, or from another point of view, deform the projective cone over C to S; Pinkham calls this construction 'sweeping out the cone' [Pinkham 1970]. Surfaces with hyperelliptic hyperplane sections were already classified by Castelnuovo, and the supernormal surfaces among them have degree 4g + 4 [Castelnuovo 1890]. They are rational ruled surfaces, and such surfaces come in two deformation types; therefore there are at least two smoothing components. This observation was the starting point of the present paper. A computer computation of the versal deformation in negative degree with Macaulay [Bayer-Stillman] gave for an example with g = 2 the number of 32 smoothing components.As the versal deformation can be chosen C * -equivariant, it makes sense to restrict to the part of negative degree. We want to show that the base space S − has 2 2g+1 one-dimensional components. First of all we have to exhibit this number of surfaces with C as hyperplane section, such that the normal bundle of C in the surface is L. The main point is that an elementary transformation on the ruled surface S in a Weierstraß point of C does not change the normal bundle of C. The composition of elementary transformations in all Weierstraß points gives an involution on S; we get 2 2g+2 /2 surfaces. This construction works for every hyperelliptic curve C and every line bundle L on C. We obtain 2 2g+1 smooth subspaces of dimension 3g of the base of the versal deformation; by a 1

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The Jacobian algebra of a graded Gorenstein singularity

Cited by 88 publications

References 22 publications

Almost split sequences and Zariski differentials

Almost split sequences and Zariski differentials

Hodge theory and deformations of affine cones of subcanonical projective varieties

Deformations of cones over hyperelliptic curves.

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