Irreducibility of a subscheme of the Hilbert scheme of complex space curves

Keem, Changho; Kim, Seonja

doi:10.1016/0021-8693(92)90190-w

Cited by 26 publications

(44 citation statements)

References 5 publications

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“…The tools that will be used for this are in essence a refinement of the technique employed by Ein (1987) and Keem and Kim (1992). The objective will be to show that there couldn't exist a component different from the one whose existence is asserted in the next proposition and the remark after it.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 98%

“…The bound in the first case is the one obtained by Ein (1987), while the bound in (a.2) is the one obtained by Keem and Kim (1992). The third bound is obtained essentially by an easy moduli counting.…”

Section: Proof Of Theorem Amentioning

confidence: 97%

“…In Ein (1986Ein ( , 1987 this has been realized by working directly with i and estimating the dimension of its tangent space, while in Keem and Kim (1992) the authors have bounded instead the dual to i component in the Picard scheme. Here we combine their methods together with a small technical lemma resolving a case that cannot be worked out by those two methods, to produce a result about d g 5 .…”

Section: Has a Unique Component Such That Its General Element Is Biramentioning

confidence: 99%

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On the Irreducibility of the Hilbert Scheme of Curves inℙ⁵

Iliev

2008

Communications in Algebra

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Denote by d g r the Hilbert scheme parametrizing smooth irreducible complex curves of degree d and genus g embedded in r . Severi (1921) claimed that d g r is irreducible if d ≥ g + r. Ein proved that the conjecture is true for r = 3 and 4, and in general that d g r is irreducible if d ≥ 2r−2 r+2 g + 2r+3 r+2 (Ein, 1986, 1987). As it is known, for r ≥ 6 the conjecture is incorrect and r = 5 remains the only unsettled case. Here I prove that d g 5 is irreducible, if d ≥ max 11 10 g + 2 g + 5 , which doesn't yet resolve Severi's conjecture for r = 5, but expands the known irreducibility range.

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Section: Introduction and Preliminary Resultsmentioning

confidence: 98%

Section: Proof Of Theorem Amentioning

confidence: 97%

Section: Has a Unique Component Such That Its General Element Is Biramentioning

confidence: 99%

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On the Irreducibility of the Hilbert Scheme of Curves inℙ⁵

Iliev

2008

Communications in Algebra

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“…It is known that H S d g is irreducible if g ≤ d − 3 (Ein, 1986), if g = d − 2 and d ≥ 7 (Keem and Kim, 1992), or if g ≤ 2d − 9 and d ≤ 11 (Guffroy, 2004). Therefore H S d 3d−18 is irreducible if 6 ≤ d ≤ 9.…”

Section: A List Of Componentsmentioning

confidence: 92%

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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“…We may therefore let C ′′ ⊂ H be a general BN-curve of degree d ′′ and genus g ′′ . Note that C ′′ passes through n general points in H, by Theorem 1.2 of [19] together with our assumption (13). Since d ′ ≥ n by (12), and the hyperplane section of a general union of rational curves of total degree d ′ is a general set of d ′ points by Corollary 1.5 of [21], we may thus let C ′ be a general union of a rational curve of degree d ′ + g ′ and −g ′ ≤ t general lines, passing through Γ.…”

Section: Proof Note Thatmentioning

confidence: 95%