Linkage of general curves of large degree

Lazarsfled, R.; Rao, Prabhakar

doi:10.1007/bfb0061648

Cited by 41 publications

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“…Recall that two curves C and C are in the same biliaison class if there is an even number of linkages connecting them. In [3] R. Lazarsfeld and P. Rao proved that curves with s(C) ≥ e(C) + 4 are minimal in their biliaison class. In [6] M. Martin-Deschamps and D. Perrin gave a construction of the minimal curves in each biliaison class starting from the minimal free graded resolution of the Rao module M (C) = H 1 * (J C ).…”

Section: Introductionmentioning

confidence: 99%

“…The notation is as in the Introduction. In order to include all possible curves in P 3 we consider a line as a minimal curve in the biliaison class of the ACM curves in P 3 .…”

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confidence: 99%

“…Let [3] or [6]); we know that P is a free graded R-module. Now let C be another curve in the same biliaison class of C. By adding a free graded R-module L we can assume that C, C have N -resolutions (not necessarily minimal) of the form…”

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confidence: 99%

“…As in [3], Section 2, Proof of Lemma 1.2, it is enough to prove that b i ≥ a i for all i = 1 · · · n. With the same notation as in [3] it is enough to prove that rank A ≥t ≥ rank B ≥t for all t ∈ Z where A ≥t = −ai≥t R(−a i ) and B ≥t = −bi≥t R(−b i ). We consider three cases: 1) t ≤ −(e(C) + 4): in this case the same proof as in [3] holds; 2) t > −s(C): in this case the same proof as in [3] holds; 3) let −s(C) ≥ t > −(e(C) + 4) and assume that there exists t such that rank A ≥t < rank B ≥t .…”

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confidence: 99%

“…We consider three cases: 1) t ≤ −(e(C) + 4): in this case the same proof as in [3] holds; 2) t > −s(C): in this case the same proof as in [3] holds; 3) let −s(C) ≥ t > −(e(C) + 4) and assume that there exists t such that rank A ≥t < rank B ≥t . We put s = −t and we have s(C) ≤ s ≤ e(C) + 3 and rank A ≥−s < rank B ≥−s .…”

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confidence: 99%

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Biliaison classes of curves in 𝐏³

Strano

2003

Proc. Amer. Math. Soc.

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Abstract. We characterize the curves in P 3 that are minimal in their biliaison class. Such curves are exactly the curves that do not admit an elementary descending biliaison. As a consequence we have that every curve in P 3 can be obtained from a minimal one by means of a finite sequence of ascending elementary biliaisons. IntroductionLet C be a curve (i.e., a locally Cohen-Macaulay subscheme equidimensional of dimension 1) in P = P 3 k over an algebraically closed field k. Let J C be its ideal sheaf in O P andIn [9] P. Rao introduced the notion of a biliaison (even linkage) class of curves and proved that two curves C and C are in the same class if and only if HBefore stating the main results of the paper we recall some basic notions concerning biliaison. A good reference for the subject is [6] or [8]. Recall that two curves C and C are in the same biliaison class if there is an even number of linkages connecting them. Definition 1. A non-ACM curve C is said to be minimal (in its biliaison class) if, for every C in the same biliaison class, HDefinition 2. Let C be a curve and let Q be a surface containing it. A curve C is said to be obtained from C by an elementary biliaison on Q of height m if there are surfaces T and V of degrees t, t + m, respectively, such that C is linked to a curve D by means of (Q, T ) and D is linked to C by means of (Q, V ).

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Section: Introductionmentioning

confidence: 99%

“…The notation is as in the Introduction. In order to include all possible curves in P 3 we consider a line as a minimal curve in the biliaison class of the ACM curves in P 3 .…”

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confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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