Subcanonical points on algebraic curves
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- by Evan M. Bullock PDF
- Trans. Amer. Math. Soc. 365 (2013), 99-122
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
- 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, DOI 10.1007/978-1-4757-5323-3
- 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, DOI 10.1007/BF02505914
- 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
- David Eisenbud and Joe Harris, Limit linear series: basic theory, Invent. Math. 85 (1986), no. 2, 337–371. MR 846932, DOI 10.1007/BF01389094
- David Eisenbud and Joe Harris, Existence, decomposition, and limits of certain Weierstrass points, Invent. Math. 87 (1987), no. 3, 495–515. MR 874034, DOI 10.1007/BF01389240
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- 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, DOI 10.1007/s00222-003-0303-x
- Brian Osserman, A limit linear series moduli scheme, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1165–1205 (English, with English and French summaries). MR 2266887, DOI 10.5802/aif.2209
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
- © Copyright 2012 Evan M. Bullock
- Journal: Trans. Amer. Math. Soc. 365 (2013), 99-122
- MSC (2010): Primary 14H55
- DOI: https://doi.org/10.1090/S0002-9947-2012-05506-1
- MathSciNet review: 2984053