On the existence of plane curves with imposed multiple points (Q5929288)

It is proved that if a projective plane curve of degree \(d\) passes with multiplicity \(m\) through \(r\) points in general position, then \(d \geq m(r-1) \prod^{r-1}_{i=2}(1- {i \over {i^2+r-1}})\). Comparing this bound with a conjecture of \textit{M. Nagata} [see Am. J. Math. 81, 766-772 (1959; Zbl 0192.13801)] and with previously known results, the author proves also that, if \(r \geq 10\), then \(d \;> m(\sqrt{r-1} - {{\pi} \over 8})\). His approach is based on a specialization of the scheme consisting of \(r\) points in general position with multiplicity \(m\) to an appropriate cluster scheme supported at a single point.