The Hilbert Scheme of Buchsbaum space curves

Kleppe, Jan O.

doi:10.5802/aif.2744

Cited by 2 publications

(2 citation statements)

References 24 publications

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“…1.2]. Semicontinuity of graded Betti numbers more generally seems to be a wellknown "folk theorem"; for example, different ideas for proofs are sketched in [40, Remark following Theorem 1.1], and in [32,Corollary 3.3]. We give a quick proof here for the sake of self-containedness.…”

Section: Secant Varieties and Border Rank Letmentioning

confidence: 99%

Apolarity and direct sum decomposability of polynomials

Buczyńska¹,

Buczyński²,

Kleppe³

et al. 2015

Michigan Math. J.

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A polynomial is a direct sum if it can be written as a sum of two non-zero polynomials in some distinct sets of variables, up to a linear change of variables. We analyse criteria for a homogeneous polynomial to be decomposable as a direct sum, in terms of the apolar ideal of the polynomial. We prove that the apolar ideal of a polynomial of degree d strictly depending on all variables has a minimal generator of degree d if and only if it is a limit of direct sums.

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Section: Secant Varieties and Border Rank Letmentioning

confidence: 99%

Apolarity and direct sum decomposability of polynomials

Buczyńska¹,

Buczyński²,

Kleppe³

et al. 2015

Michigan Math. J.

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show abstract

“…Note that the statement "a generization X ′ of X in Hilb p (P 3 ) with constant postulation" in [30,Thm. 2.8] really means "a generization X ′ of X in GradAlg(H)".…”

Section: Upgrading Of Previous Resultsmentioning

confidence: 99%

Families of artinian and low dimensional determinantal rings

Kleppe

2018

Journal of Pure and Applied Algebra

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Let GradAlg(H) be the scheme parameterizing graded quotients of R=k[x_0,...,x_n] with Hilbert function H (it is a subscheme of the Hilbert scheme of P^n if we restrict to quotients of positive dimension, see definition below). A graded quotient A=R/I of codimension c is called standard determinantal if the ideal I can be generated by the t by t minors of a homogeneous t by (t+c-1) matrix (f_{ij}). Given integers a_0\le a_1\le ...\le a_{t+c-2} and b_1\le ...\le b_t, we denote by W_s(\underline{b};\underline{a}) the stratum of GradAlg(H) of determinantal rings where f_{ij} \in R are homogeneous of degrees a_j-b_i. In this paper we extend previous results on the dimension and codimension of W_s(\underline{b};\underline{a}) in GradAlg(H) to {\it artinian determinantal rings}, and we show that GradAlg(H) is generically smooth along W_s(\underline{b};\underline{a}) under some assumptions. For zero and one dimensional determinantal schemes we generalize earlier results on these questions. As a consequence we get that the general element of a component W of the Hilbert scheme of P^n is glicci provided W contains a standard determinantal scheme satisfying some conditions. We also show how certain ghost terms disappear under deformation while other ghost terms remain and are present in the minimal resolution of a general element of GradAlg(H).Comment: Postprint replacing preprint. 29 pages. Online 26.May 2017 in Journal of Pure and Applied Algebr

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The Hilbert Scheme of Buchsbaum space curves

Cited by 2 publications

References 24 publications

Apolarity and direct sum decomposability of polynomials

Apolarity and direct sum decomposability of polynomials

Families of artinian and low dimensional determinantal rings

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