Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Subcanonical points on algebraic curves

Author: Evan M. Bullock
Journal: Trans. Amer. Math. Soc. 365 (2013), 99-122
MSC (2010): Primary 14H55
Published electronically: July 23, 2012
MathSciNet review: 2984053
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If $ C$ is a smooth, complete algebraic curve of genus $ g\geq 2$ over the complex numbers, a point $ p$ of $ C$ is subcanonical if $ K_C \cong \mathcal {O}_C\big ((2g-2)p\big )$. We study the locus $ \mathcal {G}_g\subseteq \mathcal {M}_{g,1}$ of pointed curves $ (C,p)$, where $ p$ is a subcanonical point of $ C$. Subcanonical points are Weierstrass points, and we study their associated Weierstrass gap sequences. In particular, we find the Weierstrass gap sequence at a general point of each component of $ \mathcal {G}_g$ and construct subcanonical points with other gap sequences as ramification points of certain cyclic covers and describe all possible gap sequences for $ g\leq 6$.

References [Enhancements On Off] (What's this?)

  • [ACGH85] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932 (86h:14019)
  • [BDC99] E. Ballico and A. Del Centina, Ramification points of double coverings of curves and Weierstrass points, Ann. Mat. Pura Appl. (4) 177 (1999), 293-313. MR 1747636 (2001h:14038)
  • [Cor89] Maurizio Cornalba, Moduli of curves and theta-characteristics, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 560-589. MR 1082361 (91m:14037)
  • [EH86] David Eisenbud and Joe Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), no. 2, 337-371. MR 846932 (87k:14024)
  • [EH87] David Eisenbud and Joe Harris, Existence, decomposition, and limits of certain Weierstrass points, Invent. Math. 87 (1987), no. 3, 495-515. MR 874034 (88a:14028b)
  • [Har77] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • [KZ03] Maxim Kontsevich and Anton Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), no. 3, 631-678. MR 2000471 (2005b:32030)
  • [Oss06] Brian Osserman, A limit linear series moduli scheme, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1165-1205. MR 2266887 (2007h:14042)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14H55

Retrieve articles in all journals with MSC (2010): 14H55

Additional Information

Evan M. Bullock
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005

Received by editor(s): May 3, 2010
Received by editor(s) in revised form: November 5, 2010
Published electronically: July 23, 2012
Article copyright: © Copyright 2012 Evan M. Bullock

American Mathematical Society