On the Hilbert Scheme Compactification of the Space of Twisted Cubics

Piene, Ragni; Schlessinger, Michael

doi:10.2307/2374355

Cited by 99 publications

(115 citation statements)

References 9 publications

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“…Each of these two statements implies that M 0 is smooth and rational of dimension 12 due to the results in [11] and [2]. The blowup structure of H 0 has been mentioned without proof in [2].…”

Section: The Component M 0 and Transversalitymentioning

confidence: 99%

“…The curves in H 0 \H 1 are precisely the connected locally Cohen-Macaulay curves of degree 3 in P 3 which are not contained in a plane and have Hilbert polynomial P O C (m) = 3m + 1, see e.g. [11] or [5]. Given a proper quotient O C → Q of pure dimension 1, Q is the structure sheaf O C ′ of a locally Cohen-Macaulay curve…”

Section: Stable Sheaves Supported On Cubicsmentioning

confidence: 99%

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On the moduli scheme of stable sheaves supported on cubic space curves

Freiermuth¹,

Trautmann²

2004

ajm

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Abstract. We investigate the geometry of the Simpson moduli space M P (P 3 ) of stable sheaves with Hilbert polynomial P (m) = 3m + 1. It consists of two smooth, rational components M 0 and M 1 of dimensions 12 and 13 intersecting each other transversally along an 11-dimensional, smooth, rational subvariety. The component M 0 is isomorphic to the closure of the space of twisted cubics in the Hilbert scheme Hilb P (P 3 ) and M 1 is isomorphic to the incidence variety of the relative Hilbert scheme of cubic curves contained in planes. In order to obtain the result and to classify the sheaves, we characterize M P (P 3 ) as geometric quotient of a certain matrix parameter space by a non-reductive group. We also compute the Betti numbers of the Chow groups of the moduli space.

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Section: The Component M 0 and Transversalitymentioning

confidence: 99%

Section: Stable Sheaves Supported On Cubicsmentioning

confidence: 99%

On the moduli scheme of stable sheaves supported on cubic space curves

Freiermuth¹,

Trautmann²

2004

ajm

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“…By [31], these are projectively equivalent to a curve with ideal generated by the net of quadrics x 0 (x 0 , x 1 , x 2 ) plus a cubic form q, which can be taken to be of the form q = Ax Let Y ⊆ H 3 be the locus of non-Cohen-Macaulay curves, and denote by I the 5-dimensional incidence correspondence {(p, H) ∈ P 3 × P 3 * | p ∈ H}. By the above, the quadratic part of I x for x ∈ Y gives rise to a point of I.…”

Section: Lemma 37 Let C X Be Cohen-macaulay and Let E And F Be As mentioning

confidence: 99%

“…The parameter spaces in question are suitable components of the Hilbert scheme parameterizing these curves. These components are smooth, by the work of Piene and Schlessinger [31] in the case of cubics, and Avritzer and Vainsencher [3] in the case of elliptic quartics.…”

Section: Introductionmentioning

confidence: 99%

Bott’s formula and enumerative geometry

Ellingsrud

Strømme

1996

J. Amer. Math. Soc.

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We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, Gromov-Witten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed. The results are consistent with predictions made from mirror symmetry computations. We also compute degrees of some loci in the linear system of plane curves of degrees less than 10, like those corresponding to sums of powers of linear forms, and curves carrying inscribed polygons.

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“…More is known about Hilbert schemes parametrizing particular schemes that often have small codimension (cf., e.g., [15], [8], [2], [7], [11], [12]). But, for example, it is not even known if the locus of space curves inside its Hilbert scheme is connected (cf., e.g., [5], [14]).…”

Section: Introductionmentioning

confidence: 99%

The Hilbert Scheme of Degree Two Curves and Certain Ropes

2006

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Abstract. We study families of ropes of any codimension that are supported on lines. In particular, this includes all non-reduced curves of degree two. We construct suitable smooth parameter spaces and conclude that all ropes of fixed degree and genus lie in the same component of the corresponding Hilbert scheme. We show that this component is generically smooth if the genus is small enough unless the characteristic of the ground field is two and the curves under consideration have degree two. In this case the component is non-reduced.

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On the Hilbert Scheme Compactification of the Space of Twisted Cubics

Cited by 99 publications

References 9 publications

On the moduli scheme of stable sheaves supported on cubic space curves

On the moduli scheme of stable sheaves supported on cubic space curves

Bott’s formula and enumerative geometry

The Hilbert Scheme of Degree Two Curves and Certain Ropes

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