Sections planes et multisecantes pour les courbes gauches generiques principales

Hirschowitz, Andrè

doi:10.1007/bfb0078182

Cited by 6 publications

(3 citation statements)

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“…Since Σ C is a determinantal subvariety of C e × G r d (C), it follows that for a general [C] ∈ M g , if non-empty, the scheme Σ C is equidimensional and dim(Σ C ) = ρ(g, r, d) − f (r + 1 − e + f ) + e. Note that this result does not establish the non-emptiness of Σ C , which is an issue that we will deal with in Section 3. In any event, (13) implies the dimensional estimate…”

Section: Varieties Of Secant Planes To the General Curvementioning

confidence: 99%

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Higher ramification and varieties of secant divisors on the generic curve

Farkas¹

2008

Journal of the London Mathematical Society

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For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry, and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to clarify this problem in the case of the generic curve C of given genus. We determine precisely under which conditions the cycle of e-secant k-planes is non-empty, and we compute its dimension. We also precisely determine the dimension of the variety of linear series on C carrying e-secant k-planes.

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Section: Varieties Of Secant Planes To the General Curvementioning

confidence: 99%

“…Surprisingly little is known about the validity of these classical enumerative formulas (see [13] and [15] for partial results in the case of curves in P 3 ). The aim of this paper is to clarify this problem for a general curve [C] ∈ M g .…”

Section: Introductionmentioning

confidence: 99%

Higher ramification and varieties of secant divisors on the generic curve

Farkas¹

2008

Journal of the London Mathematical Society

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“…The constant K in Corollary 1 is certainly not optimal, but the exponent d 3/2 is sharp among the curves with h 1 (N X ) = 0 (see [12], [29,Corollaire 5.18] and [20,II.3.6] for the condition h 1 (N X (−2)) = 0, [20,II.3.7] and [31] for the condition h 1 (N X (−1)) = 0, and [20,II.3.8] for the condition h 1 (N X ) = 0).…”

Section: Introductionmentioning

confidence: 99%

Maximal rank of space curves in the range A

Ballico

Ellia

Fontanari

2018

European Journal of Mathematics

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Fix positive integers d, m such that m 2 +4m+6 6 ≤ d < m 2 +4m+6 3(the so-called Range A for space curves). Let G(d, m) be the maximal genus of a smooth and connected curve, of degree d, C ⊂ P 3 such that h 0 (I C (m− 1)) = 0. Here we prove thatThe case m 2 +4m+6 4 ≤ d < m 2 +4m+6 3 was known by work of Fløystad and joint work of Ballico, Bolondi, Ellia, Mirò-Roig. To prove the case m 2 +4m+6 6 ≤ d < m 2 +4m+6 4we show that in this range for large d every integer between 0 and 1 + (m − 1)d − m+2 3 is the genus of some degree d smooth and connected curve C ⊂ P 3 such that h 0 (I C (m − 1)) = 0.and there is a smooth and connected curve C ⊂ P 3 of degree d and genus G(d, m) such that h 1 (O C (m − 2)) = 0, h i (I C (m − 1)) = 0, i = 0, 1, and h 1 (N C (−1)) = 0.In the statement of Theorem 2 we assumed that d < m 2 +4m+6 4 , because the range m 2 +4m+6 4 ≤ d < m 2 +4m+6 3 is covered by [2, Proposition 2.2 and Corollary 2.4]. In the range of [2, Corollary 2.4] our proof of Theorem 2 is very bad (it gives examples of nice curves C, but not enough to cover all d).For. Take (d, m) in the Range A and an integer g such that 0 ≤ g < G A (d, m). Is there a smooth and connected curve C ⊂ P 3 of degree d and genus g with h 0 (I C (m − 1)) = 0 ? In the upper half of Range A we know it only if G A (d, m) − g is small ([2, Proposition 4.3]). In the lower half of the Range A we adapt the proof of Theorem 2 to prove the following result. Theorem 3. Fix integers m, d, g such that m ≥ 13.8 · 10 5 , m 2 +4m+6 6 ≤ d < m 2 +4m+6 4and 0 ≤ g ≤ G A (d, m). Then there is a smooth and connected curve C ⊂ P 3 of degree d and genus g such that h 0 (I C (m − 1)) = 0 and h 1 (N C (−1)) = 0.As an easy corollary of Theorem 3 we get the following result. Corollary 1. Fix integers d, m such that m ≥ 13.8·10 5 and d ≥ m 2 +4m+6 4 . Set δ := ⌊ m 2 +4m+5 4⌋. Fix an integer g such that 0 ≤ g ≤ G A (δ, m). Then there is a smooth and connected curve C ⊂ P 3 of degree d and genus g such that h 0 (I C (m − 1)) = 0 and h 1 (N C (−1)) = 0.There are a few questions related to this paper. Fix positive integers d, m. We recall that if d > m(m − 1) (the so-called Range C) we have G(d, m) = 1 + [d(d + m 2 −4m)−r(m−r)(m−1)]/2m, where r is the only integer such that 0 ≤ r ≤ m−1 and d + r ≡ 0 (mod m), and equality holds if and only if the curve is linked to a plane curve of degree r by the complete intersection of a surface of degree m and a surface of degree d + r ([14]). Now assume

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Existence De Faisceaux Réflexifs De Rang Deux SurP 3 À Bonne Cohomologie

Hirschowitz

1987

Publications Mathématiques de L’Institut des Hautes Scientifiqu

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Existence de faisceaux réflexifs de rang deux sur P 3 à bonne cohomologie Publications mathématiques de l'I.H.É.S., tome 66 (1987), p. 105-137 © Publications mathématiques de l'I.H.É.S., 1987, tous droits réservés. L'accès aux archives de la revue « Publications mathématiques de l'I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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Sections planes et multisecantes pour les courbes gauches generiques principales

Cited by 6 publications

References 36 publications

Higher ramification and varieties of secant divisors on the generic curve

Higher ramification and varieties of secant divisors on the generic curve

Maximal rank of space curves in the range A

Existence De Faisceaux Réflexifs De Rang Deux SurP 3 À Bonne Cohomologie

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