On the Hilbert scheme of linearly normal curves in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">P</mml:mi></mml:mrow><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msup></mml:math> of relatively high degree

Ballico, Edoardo; Fontanari, Claudio; Keem, Changho

doi:10.1016/j.jpaa.2019.07.006

Cited by 10 publications

(23 citation statements)

References 17 publications

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“…and genus g = 14 (g = 15 resp.). For r = 8, a curve of degree e = 10 and genus g = 16 in P 3 -which is a (smooth) extremal curve of type (5,5) on a (smooth) quadric surface Xis embedded into P 8 by the 2-tuple embedding…”

Section: Existence and Non-existence Of Hmentioning

confidence: 99%

“…We recall that a general E ∈ G ∨ ⊂ G 3 e = G 3 10 (G is a component of G L ) induces a morphism onto a curve of type (5, 5) on a smooth quadric in P 3 . On the other hand, curves of type (5,5)…”

Section: Existence and Non-existence Of Hmentioning

confidence: 99%

“…We recall the following modified assertion of Severi, which has been given attention by some authors recently; cf. [5,22].…”

mentioning

confidence: 99%

“…Through a preliminary attempt to settle down the Modified Assertion of Severi, which is still in its infancy, one has a quite extensive knowledge about the Hilbert scheme of linearly normal curves of small index of speciality α ≤ 3 which can be summarized as follows; cf. [20,5,22].…”

mentioning

confidence: 99%

“…One obvious advantage considering the Hilbert scheme of linearly normal curves according to its index of speciality α is that the residual series of the very ample hyperplane series corresponding to the general element of a component has the fixed dimension α − 1 regardless of the values of the genus g and the degree d. Therefore one may work more effectively in exploring out several properties of the Hilbert scheme under consideration by looking at the family of curves in a fixed projective space P α−1 induced by the residual series of the hyperplane series. For one thing, in case α = 3, the residual series of hyperplane series are nets and it is possible to derive the basic properties such as irreducibility, existence as well as the number of moduli of the Hilbert scheme of linearly normal curves by considering the corresponding property of the Severi variety of plane curves, which has been proven useful to a certain extent in some previous works on the subject; [20,5,21,22]. In this paper we consider Hilbert scheme of linear normal curves with index of speciality α = 4.…”

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On the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with small index of speciality

Keem¹

2021

Preprint

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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 whose complete and very ample hyperplane linear series D have relatively small index of speciality i(D) = g − d + r. In particular we show the existence and the non-existence of certain Hilbert schemes with i(D) = 4. We also determine the irreducibility of H L 2r+4,r+8,r for 3 ≤ r ≤ 8, which are rather peculiar families in some sense.

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Section: Existence and Non-existence Of Hmentioning

confidence: 99%

Section: Existence and Non-existence Of Hmentioning

confidence: 99%

“…We recall the following modified assertion of Severi, which has been given attention by some authors recently; cf. [5,22].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with small index of speciality

Keem¹

2021

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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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On the Hilbert scheme of smooth curves of degree 15 and genus 14 in $$\mathbb {P}^5$$

Ballico,

Keem

2023

Boll Unione Mat Ital

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On the Hilbert scheme of linearly normal curves in Pr of relatively high degree

Cited by 10 publications

References 17 publications

On the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with small index of speciality

On the Hilbert scheme of linearly normal curves in $\mathbb{P}^r$ with small index of speciality

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 smooth curves of degree 15 and genus 14 in $$\mathbb {P}^5$$

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