On the existence of plane curves with prescribed multiple points

Details On the existence of plane curves with prescribed multiple points On the existence of plane curves with prescribed multiple points

1998-07-13

arxiv.org/abs/math/9807066v2

Mathematics

Algebraic Geometry

13 pages, Latex, XYpic. Revised version, to appear in Journal of Pure and Applied Algebra

Scientific paper

We address the problem of determining the degree a plane curve must have in
order to pass with multiplicity m through r points in general position. A
conjecture of Nagata states that one must have d > m \sqrt{r}. We prove the
inequalities d \geq m(r-1)\prod_{i=2}^{r-1}(1-i/(i^2+r-1)) and d > m
(\sqrt{r-1} - \pi/8).

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