Sur le nombre de composantes du schéma de Hilbert des courbes ACM de pk3

Ginouillac, Stéphane

doi:10.1016/s0764-4442(00)87488-1

Cited by 2 publications

(10 citation statements)

References 5 publications

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“…Hence we may see when C sits in the intersection of different integral components of H(d, g). There may be quite a lot of such irreducible components of H(d, g) [12]. We will soon look closely to the possible generizations of a curve of diameter one in the case β 1,c · β 2,c+4 = 0.…”

Section: With Constant Postulation and Rao Module) Whose Homogeneous mentioning

confidence: 99%

The Hilbert scheme of space curves of small diameter

Kleppe¹

2006

Ann. inst. Fourier

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This paper studies space curves C of degree d and arithmetic genus g, with homogeneous ideal I and Rao module M = H 1 * ( Ĩ), whose main results deal with curves which satisfy 0 Ext 2 R (M, M ) = 0 (e.g. of diameter, diam M ≤ 2, which means that M is non-vanishing in at most two consecutive degrees). For such curves C we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, H(d, g), at (C) under the sufficient conditions. In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of C turns out to be equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of I. We give classes of obstructed curves C for which we partially compute the equations of the singularity of H(d, g) at (C). Moreover by taking suitable deformations we show how to kill certain repeated direct free factors ("ghost-terms") in the minimal resolution of the ideal of the general curve. For Buchsbaum curves of diameter at most 2, we simplify in this way the minimal resolution further, allowing us to see when a singular point of H(d, g) sits in the intersection of several, or lies in a unique irreducible component of H(d, g). It follows that the graded Betti numbers mentioned above of a generic curve vanish, and that any irreducible component of H(d, g) is reduced (generically smooth) in the diameter 1 case.

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Section: With Constant Postulation and Rao Module) Whose Homogeneous mentioning

confidence: 99%

The Hilbert scheme of space curves of small diameter

Kleppe¹

2006

Ann. inst. Fourier

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“…Using this we take two general skew lines as in Example 2.7 and we link twice, first via a CI of type (5,2), then via a CI of type (5,4). This gives us a curve X, generic in H (12,18), with minimal resolution and a ghost term R(−5) in degree c + 3:…”

Section: The Graded Betti Numbers Of Diameter-1 Curvesmentioning

confidence: 99%

“…If we need to find a free resolution of the homogeneous ideal of a curve X linked to D, using a CI Z of type (f ′ , g ′ ) (so X and C are bilinked), we use (11) (and not (12)) and the mapping cone construction as in the big diagram above, to find a resolution of I(D)/I(Z)(f ′ + g ′ ).…”

Section: Linkagementioning

confidence: 99%

“…Thus this paper contributes to solving questions related to the number of components, irreducibility and smoothness of H(d, g), see [1], [9,10], [12,13], [23,24] for some contributions which are relevant for this paper, and [4] for a thorough study of diameter-1 curves.…”

Section: Introductionmentioning

confidence: 99%

“…v (C) = β 1,c+c(D)+4−v (D) , for v / ∈ {c + 1, c + 2, c + 3}.by(12). By (35) we get thatβ 1,c+c(D)+4−v (D) is semi-continuous, because v / ∈ {c + 1, c + 2, c + 3} is equivalent to c+c(D)+4−v / ∈ {c(D)+1, c(D)+2, c(D)+3}.…”

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The Hilbert Scheme of Buchsbaum space curves

Kleppe¹

2012

Ann. inst. Fourier

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We consider the Hilbert scheme H(d, g) of space curves C with homogeneous ideal I(C) := H 0 * (I C ) and Rao module M := H 1 * (I C ). By taking suitable generizations (deformations to a more general curve) C ′ of C, we simplify the minimal free resolution of I(C) by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of I(C ′ ). Using this for Buchsbaum curves of diameter one (M v = 0 for only one v), we establish a one-to-one correspondence between the set S of irreducible components of H(d, g) that contain (C) and a set of minimal 5-tuples that specializes in an explicit manner to a 5-tuple of certain graded Betti numbers of C related to ghost-terms. Moreover we almost completely (resp. completely) determine the graded Betti numbers of all generizations of C (resp. all generic curves of S), and we give a specific description of the singular locus of the Hilbert scheme of curves of diameter at most one. We also prove some semi-continuity results for the graded Betti numbers of any space curve under some assumptions.

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Sur le nombre de composantes du schéma de Hilbert des courbes ACM de pk3

Cited by 2 publications

References 5 publications

The Hilbert scheme of space curves of small diameter

The Hilbert scheme of space curves of small diameter

The Hilbert Scheme of Buchsbaum space curves

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