Computing Versal Deformations of Singularities with Hauser’s Algorithm

Stevens, Jan

doi:10.1007/978-3-642-39131-6_6

Cited by 4 publications

(5 citation statements)

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“…The remaining of this subsection is dedicated to make Pinkham's Theorem 3.2 explicit when S is assumed to be a non-hyperelliptic symmetric semigroup, presenting a construction made by Stoehr [St93] and . This construction can be viewed as a variant of Hauser's algorithm to compute the versal deformation space of a singularity, see [Hau83,Hau85] and [Stev13].…”

Section: A Variant Of Hauser's Algorithmmentioning

confidence: 99%

“…The main idea of the second proof is to apply a variant of Hauser's algorithm, c.f. [Hau83,Hau85] and [Stev13], deforming the affine monomial curve C S ⊂ A r instead of the associated canonical Gorenstein monomial curve in P g−1 , as required by Stoehr's original construction [St93]. All we have to do is to take the unfold of the r − 1 defining polynomials of the ideal of C S .…”

Section: Introductionmentioning

confidence: 99%

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Local Complete Intersections and Weierstrass Points

Contiero¹,

Mazzini²

2022

Preprint

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We provide a simple proof that the moduli space of complete integral Gorenstein curves with prescribed symmetric Weierstrass semigroup S is a projective space whenever the affine monomial curve associated to S is a local complete intersection and S is realizable. In addition, we also prove the same statement by assuming that the affine monomial curve is a complete intersection but not requiring that S is realizable.

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Section: A Variant Of Hauser's Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local Complete Intersections and Weierstrass Points

Contiero¹,

Mazzini²

2022

Preprint

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“…This construction can be viewed as a variant of Hauser's algorithm to compute versal deformation spaces [Hau83,Hau85] , see also [Stev13]. The standard approach in deformation theory is to successively lift infinitesimal deformations to higher order, collecting the obstructions at each stage.…”

Section: Remarkmentioning

confidence: 99%

“…This construction was given by Stoehr [St93], then improved by Contiero-Stoehr [CSt13] and generalized by Contiero-Fontes [CF18]. All these constructions can be viewed as a variant of Hauser's algorithm [Hau83,Hau85] to compute versal deformation spaces, see also [Stev13]. Next, in Section 4, we use this construction to give explicit equations for M S g,1 when S runs over the following families of symmetric semigroups, 6, 3 + 6τ, 4 + 6τ, 7 + 6τ, 8 + 6τ and 6, 1 + 6τ, 2 + 6τ, 3 + 6τ, 14 + 6τ and we show that M S g,1 is not empty in these cases by showing that the associated affine monomial curves C S can be negatively smoothable, c.f.…”

Section: Introductionmentioning

confidence: 99%

On nonnegatively graded Weierstrass points

Contiero¹,

Fontes²,

Stevens³

et al. 2021

Preprint

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We provide a new lower bound for the dimension of the moduli space of smooth pointed curves with prescribed Weierstrass semigroup at the marked point, derived from the Deligne-Greuel formula and Pinkham's equivariant deformation theory. Using Buchweitz's description of the first cohomology module of the cotangent complex for monomial curves, we show that our lower bound improves a recently one given by Pflueger. By allowing semigroups running over suitable families of symmetric semigroups of multiplicity six, we show that this new lower bound is attained, and that the corresponding moduli spaces are non-empty and of pure dimension.

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“…Given a non-hyperelliptic symmetric semigroup Ã ¤ h2; 2g C 1i, following Hauser's algorithm [14,15], and also [22], we unfold the defining equations of the associated canonically embedded monomial Gorenstein curve, introducing new variables. To take care of flatness, we explore suitable syzygies that are given by purely combinatorial arguments, see Lemma 3.6, we then obtain a compactification of M Ã g;1 by allowing Gorenstein singularities at the boundary, cf.…”

Section: Introductionmentioning

confidence: 99%