The hilbert schemes of degree three curves

Nollet, Scott

doi:10.1016/s0012-9593(97)89925-9

Cited by 38 publications

(61 citation statements)

References 9 publications

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“…Suppose that X is not contained in any quartic surface. If X is ACM, then X is linked to a line, otherwise X is linked to double lines of genus −1 by the following successive linkages: ACM X The Hilbert schemes H 2 g of double lines of genus g ≤ 0 were studied by Migliore (1986) and Nollet (1997). In the studies, the Hilbert scheme H 2 −1 was shown to be irreducible and of dimension 8.…”

Section: Nasumentioning

confidence: 99%

The Hilbert Scheme of Space Curves of Degreedand Genus 3d−18

Nasu

2008

Communications in Algebra

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Let H S d g denote the Hilbert scheme of smooth connected curves of degree d and genus g in the projective 3-space 3 . We describe all irreducible components V of H S d 3d−18 for every integer d ≤ 16. For each component V , we compute the dimension of V and determine whether H S d 3d−18 is generically smooth along V or not. We show that if d ≥ 91, then H S d 3d−18 has a unique irreducible component whose general member is contained in a smooth cubic surface, along which H S d 3d−18 is generically smooth.

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Section: Nasumentioning

confidence: 99%

The Hilbert Scheme of Space Curves of Degreedand Genus 3d−18

Nasu

2008

Communications in Algebra

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“…The results about closures also involve standard methods but are more subtle: for example to identify F sp with F 2 ∩F 3 we rely on previous work on the closure of the family of rational normal curves, such as [24,29,30,32].…”

Section: Nets Of Quadricsmentioning

confidence: 99%

Artinian Gorenstein algebras of embedding dimension four: components of PGor(H) for H=(1
Iarrobino
¹
,
Srinivasan
²

2005
Journal of Pure and Applied Algebra
23
0
0
0
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A Gorenstein sequence H is a sequence of nonnegative integers H = (1, h 1 , . . . , h j = 1) symmetric about j/2 that occurs as the Hilbert function in degrees less or equal j of a standard graded Artinian Gorenstein algebra A = R/I , where R is a polynomial ring in r variables and I is a graded ideal. The scheme PGor(H ) parametrizes all such Gorenstein algebra quotients of R having Hilbert function H and it is known to be smooth when the embedding dimension satisfies h 1 3. The authors give a structure theorem for such Gorenstein algebras of Hilbert function H = (1, 4, 7, . . .) when R = K [w, x, y, z] and I 2 wx, wy, wz (Theorems 3.7 and 3.9). They also show that any Gorenstein sequence H =(1, 4, a, . . .), a 7 satisfies the condition H j/2 is an O-sequence (Theorems 4.2 and 4.4). Using these results, they show that if H =(1, 4, 7, h, b, . . . , 1) is a Gorenstein sequence satisfying 3h − b − 17 0, then the Zariski closure C(H ) of the subscheme C(H ) ⊂ PGor(H ) parametrizing Artinian Gorenstein quotients A = R/I with I 2 wx, wy, wz is a generically smooth component of PGor(H ) (Theorem 4.6). They show that if in addition 8 h 10, then such PGor(H ) have several irreducible components (Theorem 4.9). M. Boij and others had given previous examples of certain PGor(H ) having several components in embedding dimension four or more (Pacific A. Iarrobino, H. Srinivasan / Journal of Pure and Applied Algebra 201 (2005) 62 -96 63 The proofs use properties of minimal resolutions, the smoothness of PGor(H ) for embedding dimension three (

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“…by Hartshorne [5], Bǎnicǎ and Manolache [3] and Manolache [7], [10]. One should also mention that the Hilbert scheme of (locally) Cohen-Macaulay curves in P 3 is connected for degree 3 (Nollet [15]) and degree 4 (Nollet and Schlesinger [16]). …”

Section: Introductionmentioning

confidence: 99%

Double structures on rational space curves

Vatne¹

2008

Mathematische Nachrichten

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There are very many non-reduced schemes. In this paper, we consider two examples to back this statement: we give lists of double scheme structures on a twisted cubic, and we construct rank two bundles on projective 3-space with prescribed Chern classes, from double structures on smooth rational curves. IntroductionFerrand [4] gave a procedure for making a double scheme structure on a space curve. In this paper we will illustrate this procedure by writing down explicit equations for representatives of all classes of double structures on a twisted cubic. Furthermore, we show that for any possible Chern classes of rank two bundles on P 3 , there is a bundle with these Chern classes (up to twist) with sections whose zero scheme is a double scheme structure on a smooth rational curve. Ferrand's procedure for doubling curves was extended by Bǎnicǎ and Forster [1], [2], to higher multiplicities for curves in threefolds. They described the structure of Cohen-Macaulay curves in a threefold, having smooth support, up to multiplicity 4. In particular their constructions give in principle the classification of Cohen-Macaulay curves in P 3 up to degree 4.The classification of nilpotent schemes on smooth support of any dimension (and any codimension) was done by Manolache in [8] (locally complete intersection up to multiplicity 4 and Cohen-Macaulay up to multiplicity 3) and [11] (locally complete intersection up to multiplicity 6). This was used in [9] and [12] for general classification problems. Classes of nilpotent structures (mostly of type I in a terminology inspired by [8]) were studied also in [20], [21] and [19]. The nilpotent structures were used as an essential tool to study vector bundles, e.g. by Hartshorne [5], Bǎnicǎ and Manolache [3] and Manolache [7], [10]. One should also mention that the Hilbert scheme of (locally) Cohen-Macaulay curves in P 3 is connected for degree 3 (Nollet [15]) and degree 4 (Nollet and Schlesinger [16]).

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The hilbert schemes of degree three curves

Cited by 38 publications

References 9 publications

The Hilbert Scheme of Space Curves of Degreedand Genus 3d−18

The Hilbert Scheme of Space Curves of Degreedand Genus 3d−18

Artinian Gorenstein algebras of embedding dimension four: components of PGor(H) for H=(1
Iarrobino
¹
,
Srinivasan
²

2005
Journal of Pure and Applied Algebra
23
0
0
0
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Double structures on rational space curves

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The hilbert schemes of degree three curves

Cited by 38 publications

References 9 publications

The Hilbert Scheme of Space Curves of Degreedand Genus 3d−18

The Hilbert Scheme of Space Curves of Degreedand Genus 3d−18

Artinian Gorenstein algebras of embedding dimension four: components of PGor(H) for H=(1Iarrobino1, Srinivasan2 2005Journal of Pure and Applied Algebra23000View full textAdd to dashboardCite

Double structures on rational space curves

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Artinian Gorenstein algebras of embedding dimension four: components of PGor(H) for H=(1
Iarrobino
¹
,
Srinivasan
²

2005
Journal of Pure and Applied Algebra
23
0
0
0
View full text Add to dashboard Cite