The genus of curves in p4 verifying certain flag conditions

Chiantini, Luca; Ciliberto, Ciro; Gennaro, Vincenzo Di

doi:10.1007/bf02567810

Cited by 17 publications

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“…For the existence of the flag (1.1) we refer to [CCD2], Corollary 2.8. The uniqueness follows by Bezout's Theorem and the assumption s 1 >> · · · >> s l .…”

Section: First Notice Thatmentioning

confidence: 99%

“…Since s 1 >> s 2 , by [CCD2], Lemma 2.6, we may construct on the cone C(D) over D in P r , an integral, nondegenerate, projective and arithmetically Cohen-Macaulay curve E ⊂ C(D) of degree s 1 . E lies on the cone of the flag (1.4), therefore E ∈ C(r; s 1 , s 2 , .…”

Section: First Notice Thatmentioning

confidence: 99%

“…Hence, using again [CCD2], Lemma 2.6, we may construct an arithmetically Cohen-Macaulay curve F belonging to C(r; s 1 , . .…”

Section: First Notice Thatmentioning

confidence: 99%

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Hierarchical structure of the family of curves with maximal genus verifying flag conditions

Gennaro

2007

Proc. Amer. Math. Soc.

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ABSTRACT. Fix integers r, s 1 , . . . , s l such that 1 ≤ l ≤ r − 1 and s l ≥ r − l + 1, and let C(r; s 1 , . . . , s l ) be the set of all integral, projective and nondegenerate curves C of degree s 1 in the projective space P r , such that, for all i = 2, . . . , l, C does not lie on any integral, projective and nondegenerate variety of dimension i and degree < s i . We say that a curve C satisfies the flag condition (r; s 1 , . . . , s l ) if C belongs to C(r; s 1 , . . . , s l ). Define G(r; s 1 , . . . , s l ) = max {p a (C) : C ∈ C(r; s 1 , . . . , s l )} , where p a (C) denotes the arithmetic genus of C. In the present paper, under the hypothesis s 1 >> · · · >> s l , we prove that a curve C satisfying the flag condition (r; s 1 , . . . , s l ) and of maximal arithmetic genus p a (C) = G(r; s 1 , . . . , s l ) must lie on a unique flag such as C = V 1denotes an integral projective subvariety of P r of degree s i and dimension i, such that its general linear curve section satisfies the flag condition (r − i + 1; s i , . . . , s l ) and has maximal arithmetic genus G(r − i + 1; s i , . . . , s l ). This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions. Fix integers r, d, s such that s ≥ r − 1, and let C(r; d, s) be the set of all integral, projective and nondegenerate curves of degree d in the projective space P r , not contained in any integral, projective surface of degree < s. [CCD] one proves that, when d >> s, the curves of maximal arithmetic genus in C(r; d, s) are contained in surfaces of degree s, whose general hyperplane sections are themselves curves of maximal arithmetic genus in C(r − 1; s, r − 2) (the so called "Castelnuovo curves"). In the present paper we show that this property is a particular case of a more general property.In order to state our main result, we need some preliminary notation. Fix integers r, s 1 , . . . , s l such that 1 ≤ l ≤ r − 1 and s l ≥ r − l + 1, and let C(r; s 1 , . . . , s l ) be the set of all integral, projective and nondegenerate curves C of degree s 1 in the projective space P r , such that, for all i = 2, . . . , l, C does not lie on any integral, projective and nondegenerate variety of dimension i and degree < s i . We say that a curve C satisfies the flag condition (r; s 1 , . . . , s l ) if C belongs to C(r; s 1 , . . . , s l ). Notice that C(r; s 1 , r −1)

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“…For the existence of the flag (1.1) we refer to [CCD2], Corollary 2.8. The uniqueness follows by Bezout's Theorem and the assumption s 1 >> · · · >> s l .…”

Section: First Notice Thatmentioning

confidence: 99%

Section: First Notice Thatmentioning

confidence: 99%

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Hierarchical structure of the family of curves with maximal genus verifying flag conditions

Gennaro

2007

Proc. Amer. Math. Soc.

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“…In the same vein, our approach here consists in trying to bound from below the twisted Hilbert function h 0 O C (D + nH), where |D| is the extra linear series on C. Our main results in this direction are contained in §2, while §1 is devoted to generalities and to preparatory material. In particular we note our descent lemmas 2.3 and 2.4, which play an essential role in our approach and whose proofs are consequences of previous results we proved (in collaboration with Di Gennaro) in [CCD1] and [CCD2]. As in the classical theory, this sort of descent lemma would be almost useless if not accompanied by a numerical analysis aimed to find the optimal function minimizing the twisted Hilbert functions of the curves we deal with.…”

Section: Introductionmentioning

confidence: 89%

“…As in the classical theory, this sort of descent lemma would be almost useless if not accompanied by a numerical analysis aimed to find the optimal function minimizing the twisted Hilbert functions of the curves we deal with. This numerical analysis is carried out in §3 and, as sometimes happens also in the classical case (see [CCD2]), it is the hardest part of the whole story. The subsequent sections are devoted to applications.…”

Section: Introductionmentioning

confidence: 99%

Towards a Halphen theory of linear series on curves

Chiantini

Ciliberto

1999

Trans. Amer. Math. Soc.

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Abstract. A linear series g N δ on a curve C ⊂ P 3 is primary when it does not contain the series cut by planes. For such series, we provide a lower bound for the degree δ, in terms of deg(C), g(C) and of the number s = min{i : h 0 I C (i) = 0}. Examples show that the bound is sharp. Extensions to the case of general linear series and to the case of curves in higher projective spaces are considered. IntroductionLet C be a smooth, complete curve of genus g over an algebraically closed field k of characteristic zero. If C has a line bundle L generated by global sections, of degree d with h 0 L = r + 1, i.e. a complete, base point free g r d , determining a morphism of C to P r which is birational onto its image Γ, then the classical theory of Castelnuovo provides a bound for the genus g of C in terms of d and r. Indeed, one haswhere d = m(r − 1) + with 0 ≤ ≤ r − 2 and this bound is sharp (see [H2]).A natural question in this circle of ideas is to look for a similar sharp upper bound G(r 1 , ..., r n ; d 1 , ..., d n ) for the genus of a curve C having several different linear series g ri di , i = 1, ..., n. This question has been considered by Accola in [A] and it can be solved, in some cases, with the same techniques used to prove Castelnuovo's bound (!).Another classical direction in which Castelnuovo's theory has been developed has been the search for a sharp upper bound G(r; d, s) for the genus of a nondegenerate curve of degree d in P r not lying on any irreducible surface S of degree σ < s. This is called the Halphen problem, inasmuch as it was first considered by Halphen in a famous memoir of 1882 (see [Ha], and, for more recent references, [H2] and [CCD1]). Therefore, taking into account Accola's viewpoint mentioned above, a natural development of both Castelnuovo's and Halphen's theories consists in looking for similar bounds for the genus of nondegenerate curves of degree d in P r not lying on any irreducible surface S of degree σ < s and having the further property of possessing an additional linear series g N δ , besides the g r d cut out by the hyperplanes of P r . This is the question we deal with in the present paper, which is essentially devoted to setting the foundations of this Castelnuovo-Halphen theory

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Factoriality Of Certain Hypersurfaces Of P4 With Ordinary Double Points

Ciliberto

Gennaro

2004

Encyclopaedia of Mathematical Sciences

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The genus of curves in p4 verifying certain flag conditions

Cited by 17 publications

References 6 publications

Hierarchical structure of the family of curves with maximal genus verifying flag conditions

Hierarchical structure of the family of curves with maximal genus verifying flag conditions

Towards a Halphen theory of linear series on curves

Factoriality Of Certain Hypersurfaces Of P4 With Ordinary Double Points

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