An asymptotic existence theorem for plane curves with prescribed singularities

Details An asymptotic existence theorem for plane curves with prescribed singularities An asymptotic existence theorem for plane curves with prescribed singularities

1999-04-19

arxiv.org/abs/math/9904097v2

Mathematics

Algebraic Geometry

16 pages, Latex; corrections in sections 3 and 5, misprints removed

Scientific paper

Let $d,m_1,...,m_r$ be ($r+1$) positive integers, and $P_1,...,P_r$ be $r$ general points in the projective plane ; let $m$ be a positive integer. We prove that there exists a bound $d_0(m)$ such that : If $m_i < m$ ($0 d_0(m)$ then the linear system $L$ of plane curves of degree $d$ having a multiplicity at least $m_i$ at each point $P_i$ has the expected dimension ; moreover, if $L$ is not empty, there exists an irreducible plane curve of degree $d$, smooth away from the $r$ points $P_i$, and having an ordinary singularity of the prescribed multiplicity $m_i$ at each point $P_i$. This curve may be isolated in $L$.

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