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Towards a Halphen theory of linear series on curves. (English) Zbl 0930.14020

This paper deals with the question of looking for a sharp bound for the genus of non-degenerate curves of degree \(d\) in \(\mathbb{P}^r\) (over an algebraically closed field of characteristic 0), not lying on any irreducible surface \(S\) of degree less than a fixed \(s\) and having the further property of possessing an additional linear series \(g^N_\delta\), besides the \(g^r_d\) cut out by the hyperplanes of \(\mathbb{P}^r\). The authors call this point of view Castelnuovo-Halphen theory and their paper sets the foundations of this subject and shows the first applications to some classes of curves.
In the introduction the general problem is posed in the classical context (and as a generalization) of both Castelnuovo’s and Halphen’s theories and many references are discussed to orientate the reader inside the subject. – Section 1 is devoted to generalities and preliminary material. Both classical theories are based on the idea of bounding from below the Hilbert function of an irreducible, non-degenerate curve \(\Gamma\) in \(\mathbb{P}^r\) (or in bounding from below the function \(h^0(O_C(nH))\), where \(C\) is the normalization of \(\Gamma\) and \(H\) is the pull-back to \(C\) of a hyperplane in \(\mathbb{P}^r)\). Analogously the approach of this paper (see section 2) consists in bounding from below the twisted Hilbert function \(h^0(O_c(D+nH))\), where \(|D|\) is an extra linear series on \(C\). In particular two descent lemmas (2.3 and 2.4) are proved as consequences of results of both authors with V. Di Gennaro in previous papers [L. Chiantini, C. Ciliberto and V. Di Gennaro, Duke Math. J. 70, No. 2, 229-245 (1993; Zbl 0799.14011) and Manuscr. Math. 88, No. 1, 119-134 (1995; Zbl 0866.14016)].
In order to be able to apply effectively the results of section 2 a numerical analysis is carried out in section 3 tofind the optimal function minimizing the twisted Hilbert functions of the curves they deal with. Section 4 is devoted to apply the previous results (especially 2.3) to curves in \(\mathbb{P}^3\). In particular the authors prove, in terms of \(d,g\) and \(s\), a sharp lower bound for the degree \(\delta\) of \(g^N_\delta\) on a curve of degree \(d\) and genus \(g\), not lying on a surface of degrees less than \(s\) in \(\mathbb{P}^3\), under the additional hypothesis that the series does not contain the series cut out by the planes (4.1 and 4.4). – Section 5 is devoted to applications to curves in higher dimensional projective spaces, where only partially sharp bounds are given; in particular a lower bound for the degree of \(g^N_\delta\) with \(N\geq 2\) and another proof of R. D. M. Accola’s bound for the gonality of Castelnuovo’s curves [Trans. Am. Math. Soc. 251, 357-373 (1979; Zbl 0417.14021)].

MSC:

14H50 Plane and space curves
14C20 Divisors, linear systems, invertible sheaves
14H45 Special algebraic curves and curves of low genus
14N05 Projective techniques in algebraic geometry
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