On a theorem of Castelnuovo, and the equations defining space curves

Gruson, Laurent; Lazarsfeld, Robert; Peskine, Christian

doi:10.1007/bf01398398

Cited by 283 publications

(258 citation statements)

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“…These results depend on the regularity theorem for space curves of Gruson, Lazarsfeld and Peskine in [8]. Let ω denote the natural map ∧ n−2 H 0 (N C|X ) −→ H 0 (∧ n−2 N C|X ).…”

Section: Nonvanishing Results For K X Nefmentioning

confidence: 99%

“…Since (4.1) does not split there is a point p 1 in the zero locus ofs such that s is not zero in the geometric fiber of ( Thus to complete the proof of the theorem we need to show that it is possible to pick G ∈ S k such that G vanishes at every p j , except p 1 and does not vanish at p 1 . Since C is smooth of degree d ≤ k + 2, then by the main result of [8] the natural map…”

Section: Theorem 41mentioning

confidence: 99%

“…In the sections that follow several applications of this technique are given. By appealing to a regularity result for space curves in [8], this technique is used to prove some nonvanishing results for the infinitesimal Abel-Jacobi mapping Φ :…”

Section: Introductionmentioning

confidence: 99%

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Computing the infinitesimal invariants associated to deformations of subvarieties

Westhoff¹

1998

Pacific J. Math.

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The purpose of this article is to study and describe a method for computing the infinitesimal invariants associated to deformations of subvarieties. An interpretation of the infinitesimal invariant of normal functions as a pairing similar to the infinitesimal Abel-Jacobi mapping is given. The computation of both invariants for certain forms is then reduced to a residue computation at a finite number of points of the subvariety. Applications of this technique include a nonvanishing result for the infinitesimal Abel-Jacobi mapping leading to finiteness results for low degree rational curves on complete intersection threefolds with trivial canonical bundle and a generalization of a formula of Voisin for the infinitesimal invariant of certain normal functions.

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Section: Nonvanishing Results For K X Nefmentioning

confidence: 99%

Section: Theorem 41mentioning

confidence: 99%

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Computing the infinitesimal invariants associated to deformations of subvarieties

Westhoff¹

1998

Pacific J. Math.

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“…As for part (2), note that deg C = s i=1 deg C i . Since the Eisenbud-Goto conjecture holds for integral curves [GLP83], we have reg…”

Section: From Curves To Graphsmentioning

confidence: 99%

Regulating Hartshorne’s connectedness theorem

2017

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A classical theorem by Hartshorne states that the dual graph of any arithmetically CohenMacaulay projective scheme is connected. We give a quantitative version of Hartshorne's result, in terms of Castelnuovo-Mumford regularity. If X ⊂ P n is an arithmetically Gorenstein projective scheme of regularity r + 1, and if every irreducible component of X has regularity ≤ r , we show that the dual graph of X is r+r −1 r -connected. The bound is sharp. We also provide a strong converse to Hartshorne's result: Every connected graph is the dual graph of a suitable arithmetically Cohen-Macaulay projective curve of regularity ≤ 3, whose components are all rational normal curves. The regularity bound is smallest possible in general.Further consequences of our work are: (1) Any graph is the Hochster-Huneke graph of a complete equidimensional local ring. (This answers a question by Sather-Wagstaff and Spiroff.) (2) The regularity of a curve is not larger than the sum of the regularities of its primary components.

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“…(This is easy for 1 -dimensional primes. It was proved for 2-dimensional primes representing smooth curves by Castelnuovo, and for arbitrary 2-dimensional primes by Gruson et al (1983). Various weaker results are known for higher dimensional primes; see for example Lazarsfeld (1987).…”

Section: ) Regularity Boundsmentioning

confidence: 99%