Completeness of normal rational curves
A \(k\)-arc in \(\mathrm{PG}(n,q)\), the \(n\)-dimensional projective space over \(\mathrm{GF}(q)\), is a set \(K\) of \(k\) points of \(\mathrm{PG}(n,q)\), \(k\geq n+1\), with the property that no \(n+1\) points of \(K\) lie in a hyperplane. A normal rational curve in \(\mathrm{PG}(n,q)\), \(2\leq n\leq q-2\), is a \((q+1)\)-arc that is projectively equivalent to the \((q+1)\)-arc given, in homogeneous coordinates, by \(\{(1,t,\dots,t^ n)\); \(t\in \mathrm{GF}(q)\}\cup\{(0,\dots,1)\}\). In \(\mathrm{PG}(q-1,q)\), all \((q+1)\)-arcs are called normal rational curves. The author investigates whether a normal rational curve of \(\mathrm{PG}(n,q)\) can be extended to a \((q+2)\)-arc. He improves previous results of Storme and Thas, using a result by Kovács. The problem is now completely solved for large prime numbers \(q\) and odd nonsquare prime powers \(q=p^{2h+1}\) with \(p\) prime, \(p\ge p_0(h)\), \(h\ge 1\), where \(p_0(h)\) is an odd prime number which depends on \(h\).
scientific article; zbMATH DE number 130430
scientific article