Subcanonical points on algebraic curves

Bullock, Evan

doi:10.1090/S0002-9947-2012-05506-1

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 $g\leq 6$.

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