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Secant spaces and Clifford’s theorem. (English) Zbl 0741.14035
From the introduction: The following theorem A is basic for the results of this paper:
Any reduced irreducible non-degenerate and linearly normal curve \(C\) of degree \(d\geq 4r-7\) in \(\mathbb{P}^ r (r\geq 2)\) has a \((2r-3)\)-secant \((r- 2)\)-plane.
This theorem is a special case of a more general theorem which we prove in the first part of this paper. By examples, we will show that the bound on the degree of \(C\) seems to be the best possible bound only for \(r\leq 4\). — In the second part we first use theorem A to clarify the relation between two invariants of a smooth, irreducible projective curve \(C\) of genus \(g\geq 4\): the gonality \(k\) of \(C\) and the Clifford index \(c\) of \(C\). In particular, we recover E. Ballico’s result [Proc. Am. Math. Soc. 97, 217-218 (1986; Zbl 0591.14020)] that every possible value of the Clifford index of a curve of given genus really occurs. — An application of theorem A is an improvement of Clifford’s classical theorem B (“refined Clifford”):
On a \(k\)-gonal curve \(C (k\geq 3)\) of genus \(g\) any \(g^ r_ d\) of degree \(k-3\leq d\leq 2g-2-(k-3)\) satisfies \(2r\leq d-(k-3)\).
In part 3 of this paper we use theorem A to determine the maximal degree of all linear systems of degree \(d\leq g-1\) on \(C\) which compute the Clifford index \(c\) of \(C\) (we say that \(g^ r_ d\) computes \(c\) if \(d\leq g-1\) and \(d-2r=c)\). Our main result is theorem C:
Any \(g^ r_ d\) \((d\leq g-1)\) on \(C\) computing \(c\) has degree \(d\leq 2(c+2)\) unless \(C\) is hyperelliptic or bi-elliptic.

MSC:
14N05 Projective techniques in algebraic geometry
14H45 Special algebraic curves and curves of low genus
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
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