Variety of Special Nets of Degree g-1 on Double Coverings of a Smooth Plane Quartic of Genus 9

  • Kyung-Hye Cho
  • Changho Keem
  • Akira Ohbuchi
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 8)


Let C be a complete non-singular curve over ℂ of genus g. We denote by W d r (C) the subscheme of the Picard variety Pic d (C) whose support is the locus of complete linear series of degree d and dimension at least r. In case d > g + r - 1, W d r (C) = Pic d (C) and if d = g + r - 1, W d r (C) has dimension d. Therefore the dimension of W d r (C) is independent of C in the range dg + r - 1. If dg + r - 2, one knows that dimW d r (C) ≥ p(d, g, r):= g - (r + 1) (g - d + r) for any curve C and is equal to p(d, g, r) for general curve C (see Kleiman-Laksov [9] and Griffiths-Harris [6]). But the dimension of W d r (C) might be greater than p(d, g, r) for some special curve C. Moreover, for curves C with dim W d r (C) > p(d, g, r), C must be of some special type of curves. The first important result along this line is the following well-known theorem of H. Martens which has been extended by a theorem of D. Mumford.


Singular Point Exact Sequence Irreducible Component Double Covering Algebraic Curve 
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Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Kyung-Hye Cho
    • 1
  • Changho Keem
    • 1
  • Akira Ohbuchi
    • 2
  1. 1.Department of MathematicsSeoul National UniversitySeoulKorea
  2. 2.Department of Mathematics, Faculty of Integrated Arts and SciencesTokushima UniversityTokushimaJapan

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