Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE NUMBER OF LINEAR SYSTEMS COMPUTING THE GONALITY

  • Coppens, Marc (Department Industrieel Ingenieur en Biotechniek Campus)
  • Published : 2000.05.01
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Abstract

Let C be a smooth k-gonal curve of genus g. We study the number of pencils of degree k on C. In case $g\geqk(k-a)/2$ we state a conjecture based on a discussion on plane models for C. From previous work it is known that if C possesses a large number of pencils then C has a special plane model. From this observation the conjectures are split up in two cases : the existence of some types of plane curves should imply the existence of curves C with a given number of pencils; the non-existence of plane curves should imply the non-existence of curves C with some given large number of pencils. The non-existence part only occurs in the range $k(k-1)/2\leqg\leqk(k-1)/2] if k\geq7$. In this range we prove the existence part of the conjecture and we also prove some non-existence statements. Those result imply the conjecture in that range for $k\leq10$. The cases $k\leq6$ are handled separately.

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References

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