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Completeness and non-speciality of linear series on space curves and deficiency of the corresponding linear series on normalization. (English) Zbl 0599.14004

Curves Semin. Queen’s, Vol. 3, Kingston/Can. 1983, Queen’s Pap. Pure Appl. Math. 67, Exp. J, 18 p. (1984).
[For the entire collection see Zbl 0584.00015.]
Let us consider a reduced, irreducible, non-degenerate curve \(\Gamma\) in \({\mathbb{P}}^ r_ k\) (k an algebraically closed field). Let \(A=\oplus_{n\geq 0}A_ n\quad be\) a saturated, homogeneous ideal of \(k[x_ 0,...,x_ r]\) which strictly contains the ideal of \(\Gamma\). Denote by \(\Delta\) (A) the effective divisor defined by A on the normalization C of \(\Gamma\) and by D the divisor on C defined by the hyperplane section. One has the canonical maps \(\sigma_ n:\quad A_ n\to H^ 0(C,{\mathcal O}_ C(nD-\Delta (A))\) and denotes \(\lambda_ A^{(n)}=\dim Co\ker \sigma_ n\); thus \(\lambda_ A^{(n)}\) is the deficiency of the linear system cut out on C by the hypersurfaces of \(A_ n\) outside \(\Delta\) (A). When \(n\gg 0\), \(\lambda_ A^{(n)}\) is independent of n and can be described in geometric terms (a previous result of the author). In this paper the author gives computable bounds for such n, with special attention to special ideals A (adjoint ideals) or special classes of curves \(\Gamma\).
Reviewer: M.Stoia

MSC:

14C20 Divisors, linear systems, invertible sheaves
14H45 Special algebraic curves and curves of low genus

Citations:

Zbl 0584.00015