The genus of curves in P4 and P5

Rathmann, Jürgen

doi:10.1007/bf01221588

Math Z

1989

DOI: 10.1007/bf01221588

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The genus of curves in P4 and P5

Jürgen Rathmann¹

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Cited by 7 publications

(4 citation statements)

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“…in P 4 (P 5 resp.) follows from the work of Rathman [32]. However, we will show the existence and the linear normality of such curves through explicit examples we have seen in this proof.…”

mentioning

confidence: 82%

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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“…in P 4 (P 5 resp.) follows from the work of Rathman [32]. However, we will show the existence and the linear normality of such curves through explicit examples we have seen in this proof.…”

mentioning

confidence: 82%

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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“…It is, however, not so difficult to prove existence. One may, as Dolcetti and Pareschi do in [6], prove existence using Rathmann's work [37]. By [37] one knows that there exists a smooth connected curve contained in a smooth del Pezzo surface in P 4 , of degree d > 4 and genus g provided…”

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confidence: 97%

The Hilbert scheme of space curves sitting on a smooth surface containing a line

Kleppe

2016

Rend. Circ. Mat. Palermo, II. Ser

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We continue the study of maximal families W of the Hilbert scheme, H(d, g) sc , of smooth connected space curves whose general curve C lies on a smooth degree-s surface S containing a line. For s ≥ 4, we extend the two ranges where W is a unique irreducible (resp. generically smooth) component of H(d, g) sc . In another range, close to the boarder of the nef cone, we describe for s = 4 and 5 components W that are non-reduced, leaving open the non-reducedness of only 3 (resp. 2) families for s ≥ 6 (resp. s = 5), thus making progress to recent results of Kleppe and Ottem in [28]. For s = 3 we slightly extend previous results on a conjecture of non-reduced components, and in addition we show its existence in a subrange of the conjectured range.

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“…Given a triple (𝑟, 𝑑, 𝑔), find an integral and non-degenerate curve 𝑋 ⊂ P 𝑟 such that deg(𝑋) = 𝑑 and 𝑝 𝑎 (𝑋) = 𝑔. This is known only for 𝑟 = 3 by Gruson and Peskine [35,39] and known, except for very large 𝑔, in P 4 and P 5 [64]. Refined versions, prescribing that the curve is linearly normal appear in [49,61,62].…”

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confidence: 99%

“…For 𝑟 = 3, see [35,39,62]. For 𝑟 = 4, 5 see [61,64]. The query about 𝐴 1 (𝑟, 𝑑, 𝑔) was raised by R. Hartshorne [41,Prob 4d.4].…”

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confidence: 99%

Curves in projective spaces: questions and remarks

Ballico

2022

Proyecciones (Antofagasta)

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The genus of curves in P4 and P5

Cited by 7 publications

References 5 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

The Hilbert scheme of space curves sitting on a smooth surface containing a line

Curves in projective spaces: questions and remarks

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