Number of singular points of a genus $g$ curve with one point at infinity

Details Number of singular points of a genus $g$ curve with one point at infinity Number of singular points of a genus $g$ curve with one point at infinity

2009-08-31

arxiv.org/abs/0908.4500v1

Mathematics

Algebraic Geometry

Scientific paper

We bound the maximal number N of singular points of a plane algebraic curve C
that has precisely one place at infinity with one branch in terms of its first
Betti number $b_1(C)$. Asymptotically we prove that $N<\sim{17/11}b_1(C)$ for
large $b_1$. In particular, in the case of curves with one place at infinity,
we confirm the Zaidenberg and Lin conjecture stating that $N\le 2b_1+1$.

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