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Linear series on 4-gonal curves. (English) Zbl 0972.14021

From the introduction: Recall that the gonality of a curve is the minimal degree of a linear pencil on it thus measuring the deviation from rationality of the curve in a different way than the genus of the curve does. The gonality provides a stratification of the moduli space \(M_g\) of curves of genus \(g\). It seems natural to ask if/how the linear pencils of degree \(k\) may be used to describe the “non-trivial” linear series on curves in \(M_g\) of gonality \(k\). At least for the general curve of genus \(g\) and gonality \(k\) an amenable answer would help to establish a Brill-Noether theory for such curves.
In this paper we study this question for curves of low gonality – especially for 4-gonal curves.
After passing through (and motivated by) the well-known theory of linear series on 2- and 3-gonal curves we will define in §1 certain types of (complete) linear series on a \(k\)-gonal curve which are well-adopted to identify and non-trivial linear series on it in the case \(k=4\). In §2 we apply this typification to investigate the minimal degree \(s(r)\) of birational models of the \(k\)-gonal curve in \(\mathbb{P}^r\) \((r\geq 2)\). Again, the results are complete only in case \(k=4\), for the general 4-gonal curve of genus \(g\); in that case \(s(r)=[{g+ 4r-1\over 2}]\) if \(g\) is not too small.
In §3 we single out irreducible components of the varieties \(W_d^r(C)\) \((d<g,\;r\geq 1)\) of special divisors on the general 4-gonal curve \(C\) obtaining for nets (i.e., \(r=2)\) a fairly complete picture. In particular, for \(d\geq s(r)\) the variety \(W_d^r(C)\) is not equi-dimensional provided that \(g\geq 4(r+1)\) \((r\geq 2)\).

MSC:

14H51 Special divisors on curves (gonality, Brill-Noether theory)
14C20 Divisors, linear systems, invertible sheaves
14H45 Special algebraic curves and curves of low genus
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