The Hilbert Scheme of Space Curves of Degree<i>d</i>and Genus 3<i>d</i>−18

Nasu, Hirokazu

doi:10.1080/00927870802175089

Cited by 2 publications

(3 citation statements)

References 21 publications

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“…Proof. The proof is based on the proofs of similar statements in [9] and [30] for the case n = 2 and curves on cubic surfaces in P 3 . One first shows that there is a flat family whose special fibre is a cone with a general fibre a rational normal surface scroll as follows.…”

Section: Appendix a Specializationmentioning

confidence: 99%

On the Hilbert scheme of linearly normal curves in $$\mathbb {P}^4$$ of degree $$d = g+1$$ and genus g

Keem

Kim

2019

Arch. Math.

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We study the Hilbert scheme of smooth, irreducible, nondegenerate and linearly normal curves of degree d and genus g in P r (r ≥ 3) whose complete and very ample hyperplane linear series D have relatively small index of speciality i(D) = g − d + r. In particular we completely determine the existence as well as the non-existence of Hilbert schemes of linearly normal curves H L d,g,r for every possible triples (d, g, r) with i(D) = 5 and r ≥ 3. We also determine the irreducibility of the Hilbert scheme H L g+r−5,g,r when the genus g is near to the minimal possible value with respect to the dimension of the projective space P r for which H L g+r−5,g,r = ∅, say r + 9 ≤ g ≤ r + 11. In the course of proofs of key results, we show the existence of linearly normal curves of degree d ≥ g + 1 with arbitrarily given index of speciality with some mild restriction on the genus g.

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Section: Appendix a Specializationmentioning

confidence: 99%

On the Hilbert scheme of linearly normal curves in $$\mathbb {P}^4$$ of degree $$d = g+1$$ and genus g

Keem

Kim

2019

Arch. Math.

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“…For (14,11), we now sketch a proof, which was explained to us by H. Nasu, using similar techniques as in [25].…”

Section: Linkagementioning

confidence: 99%

“…Indeed, start from a smooth curve C ∈ H S 2,5 contained in a smooth quadric Q : if we identify Q to P 1 × P 1 , C is a curve of bidegree (2,3). One can show that such a curve is 4-regular, and applying a theorem of Martin-Deschamps and Perrin [25,Theorem 3.4], we obtain that a general linkage of type [4,4] yields a smooth curve C ∈ H S 14,11 . Denote by W the family of curves in H S 14,11 obtained by this process.…”

Section: Linkagementioning

confidence: 99%

Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

Blanc

Lamy

2012

Proceedings of the London Mathematical Society

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We characterize smooth curves in P 3 whose blow-up produces a threefold with anticanonical divisor big and nef.These are curves C of degree d and genus g lying on a smooth quartic, such that (i) 4d − 30 g 14 or (g, d) = (19, 12), (ii) there is no 5-secant line, 9-secant conic nor 13-secant twisted cubic to C. This generalizes the classical similar situation for the blow-up of points in P 2 .We describe then Sarkisov links constructed from these blow-ups, and are able to prove the existence of Sarkisov links which were previously only known as numerical possibilities.We obtain in fact a more precise result, which is our main theorem below. Here, we denote by H S g,d the Hilbert scheme of smooth irreducible curves of genus g and degree d in P 3 . We always work over the base field C of complex numbers.Theorem 1.1 (see Table 1). Let C ∈ H S g,d be a curve of genus g and degree d. We denote by X the blow-up of P 3 along C and by −K X its anticanonical divisor. Consider the sets A 0These pairs form a partition of all (g, d) corresponding to non-empty H S g,d with 4d − 30 g 14 or (g, d) = (19, 12), and we have the following characterizations. or (g, d) ∈ A 2 and there is no 4-secant line to C;(ii) The variety X is weak Fano, that is, −K X is big and nef., if and only if one of the following condition holds:and there is no 4-secant line to C; (c) (g, d) ∈ A 3 , there is no 5-secant line to C, and C is contained in a smooth quartic; (d) (g, d) ∈ A 4 , there is no 5-secant line, 9-secant conic, nor 13-secant twisted cubic to C, and C is contained in a smooth quartic. Moreover, Condition (i) corresponds to a non-empty open subset of H S g,d if (g, d) ∈ A 2 , and Condition (ii) corresponds to a non-empty open subset of H S g,d if (g, d) ∈ A 3 ∪ A 4 . Furthermore, for a general curve in these sets, there are finitely many irreducible curves intersecting trivially K X , except for (g, d) ∈

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

Cited by 2 publications

References 21 publications

On the Hilbert scheme of linearly normal curves in $$\mathbb {P}^4$$ of degree $$d = g+1$$ and genus g

On the Hilbert scheme of linearly normal curves in $$\mathbb {P}^4$$ of degree $$d = g+1$$ and genus g

Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

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