Nonsingular curves passing through points of \({\mathbb{P}}^ 2\) in generic position. I
The paper investigates the following question: given n points in \({\mathbb{P}}^ 2\), subject to some condition of ''genericity'', what is the least integer \(\delta\) such that there is a nonsingular curve of degree less or equal to \(\delta\) passing through the points? An evaluation for \(\delta\) is given when no three of the points are on a line. The value of \(\delta\) is also obtained if different notions of general position for n points are considered [see for instance \textit{A. V. Geramita} and \textit{F. Orecchia}, J. Algebra 70, 116-140 (1981; Zbl 0464.14007)]. Special attention is paid to points in uniform position: it is shown that if \(n=\left( \begin{matrix} d+2\\ 2\end{matrix} \right)+h\), \(0\leq h\leq 2\) and n points are in uniform position, then \(\delta =d+1\). Analogous results are proved for other ranges of h. Many examples are also discussed. [For part II see the following review.]
scientific article; zbMATH DE number 3886037