Some applications of a classical method of Castelnuovo. (Italian) Zbl 0612.14028

It is a classical result due to G. Castelnuovo [Rend. Circ. Mat. Palermo 7, 89-110 (1893), see also Memorie scelte (1973)] that the arithmetic genus of an irreducible, non degenerate curve of degree \(d\) in a projective space \({\mathbb{P}}^ r\) (over an algebraically closed field of characteristic zero) has an upper bound which is a function of d and r. The original idea of Castelnuovo, essentially based on the so called ”uniform position lemma”, is briefly reviewed in this paper and some other applications of his method are given in order to prove a few uniqueness theorems for certain linear series on some classes of curves, like curves of high genus with respect to the degree and the dimension of the projection space in which they lie, subcanonical curves, complete intersection curves. In particular a proof is given of a result stated, but never proved, by M. Noether [”Zur Grundlegung der Theorie der algebraischen Raumkurven”, Math.-Phys. Abhdl., Kgl. Preuss. Akad. Wiss. Berlin (1883)] which gives a maximum for the dimension of linear series of given degree on a smooth plane curve of degree d\(>3\), and a characterization of the linear series for which the maximum is attained. This theorem has been recently proved in a different way, and in a more general form, by R. Hartshorne [J. Math. Kyoto Univ. 26, 375-386 (1986; Zbl 0613.14008)]. Results related to the present paper are also contained in the note by the author and R. Lazarsfeld in Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lect. Notes Math. 1092, 198-213 (1984; Zbl 0548.14016).


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