Refining Castelnuovo-Halphen bounds

Details Refining Castelnuovo-Halphen bounds Refining Castelnuovo-Halphen bounds

2011-07-19

arxiv.org/abs/1107.3672v1

Mathematics

Algebraic Geometry

18 pages

Scientific paper

Fix integers $r,d,s,\pi$ with $r\geq 4$, $d\gg s$, $r-1\leq s \leq 2r-4$, and $\pi\geq 0$. Refining classical results for the genus of a projective curve, we exhibit a sharp upper bound for the arithmetic genus $p_a(C)$ of an integral projective curve $C\subset {\mathbb{P}^r}$ of degree $d$, assuming that $C$ is not contained in any surface of degree $ \pi$. Next we discuss other types of bound for $p_a(C)$, involving conditions on the entire Hilbert polynomial of the integral surfaces on which $C$ may lie.

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