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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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References:
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