Deforming Stanley–Reisner schemes

Altmann, Klaus; Christophersen, Jan Arthur

doi:10.1007/s00208-010-0490-x

Cited by 18 publications

(31 citation statements)

References 16 publications

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“…The pairs (q, p) are all pairs such that the meet of q and p is the parent of p. They, together with u ∅,ρ correspond to the minimal generators of the first cotangent module T 1 (k[x [2]×P ]/L(2, P )) by Corollary 6.11. For an element p ∈ P let b 1 , .…”

Section: 1mentioning

confidence: 97%

“…Let us call the part of the above ladder starting from the column with b 1 2 , the left leg. The minors of the left leg are precisely the equations (2). The minors formed by taking the first column (only the two lowest entries) and a column of the left leg, gives the equations (4).…”

Section: Examplesmentioning

confidence: 99%

“…Proof. We apply the local criterion of flatness [ [2]×P /L(2, P )) for any quadratic letterplace ideal of a poset P . As we shall see the elements of this module are remarkably simple in form.…”

Section: 2mentioning

confidence: 99%

“…ranging over all pairs (q, p) such that the meet of q and p is the parent of p. This will be the base ring for our family of deformations. Let B(2, P ) be the ring B ⊗ k k[x [2]×P ]. This is the ring where the ideal of the full deformation family lives.…”

Section: Introductionmentioning

confidence: 99%

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Rigid ideals by deforming quadratic letterplace ideals

Fløystad

Nematbakhsh

2018

Journal of Algebra

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We compute the deformation space of quadratic letterplace ideals L(2, P ) of finite posets P when its Hasse diagram is a rooted tree. These deformations are unobstructed. The deformed family has a polynomial ring as the base ring. The ideal J(2, P ) defining the full family of deformations is a rigid ideal and we compute it explicitly. In simple example cases J(2, P ) is the ideal of maximal minors of a generic matrix, the Pfaffians of a skew-symmetric matrix, and a ladder determinantal ideal.Date: September 21, 2018.

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Section: 1mentioning

confidence: 97%

Section: Examplesmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rigid ideals by deforming quadratic letterplace ideals

Fløystad

Nematbakhsh

2018

Journal of Algebra

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“…Scaffolding used to construct[3][4] It is easily verified that this map is homogeneous, and that φ : Pic(Y S ) → Pic(Y ) is given Hence the stability condition ω = (2, 1), is mapped into the wall spanned by(1, 0, 0) and (1, 1, 1). Let Y be the toric variety defined by weight matrix M and stability condition (2, 1, 1).…”

mentioning

confidence: 99%

From cracked polytopes to Fano threefolds

Prince

2020

manuscripta math.

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We construct Fano threefolds with very ample anti-canonical bundle and Picard rank greater than one from cracked polytopes -polytopes whose intersection with a complete fan forms a set of unimodular polytopes -using Laurent inversion; a method developed jointly with Coates-Kasprzyk. We also give constructions of rank one Fano threefolds from cracked polytopes, following work of Christophersen-Ilten and Galkin. We explore the problem of classifying polytopes cracked along a given fan in three dimensions, and classify the unimodular polytopes which can occur as 'pieces' of a cracked polytope.

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Toric Rings, Inseparability and Rigidity

Bigdeli

Herzog

2018

Springer Proceedings in Mathematics &Amp; Statistics

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This article provides the basic algebraic background on infinitesimal deformations and presents the proof of the well-known fact that the non-trivial infinitesimal deformations of a K-algebra R are parameterized by the elements of cotangent module T 1 (R) of R. In this article we focus on deformations of toric rings, and give an explicit description of T 1 (R) in the case that R is a toric ring. In particular, we are interested in unobstructed deformations which preserve the toric structure. Such deformations we call separations. Toric rings which do not admit any separation are called inseparable. We apply the theory to the edge ring of a finite graph. The coordinate ring of a convex polyomino may be viewed as the edge ring of a special class of bipartite graphs. It is shown that the coordinate ring of any convex polyomino is inseparable. We introduce the concept of semi-rigidity, and give a combinatorial description of the graphs whose edge ring is semi-rigid. The results are applied to show that for m − k = k = 3, G k,m−k is not rigid while for m − k ≥ k ≥ 4, G k,m−k is rigid. Here G k,m−k is the complete bipartite graph K m−k,k with one edge removed.

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Deforming Stanley–Reisner schemes

Abstract: We study the deformation theory of projective Stanley-Reisner schemes associated to combinatorial manifolds. We achieve detailed descriptions of first order deformations and obstruction spaces. Versal base spaces are given for certain StanleyReisner surfaces.

Cited by 18 publications

References 16 publications

Rigid ideals by deforming quadratic letterplace ideals

Rigid ideals by deforming quadratic letterplace ideals

From cracked polytopes to Fano threefolds

Toric Rings, Inseparability and Rigidity

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