\(k\)-arcs and dual \(k\)-arcs (Q1322292)

In \(\text{PG}(n,q)\), a \(k\)-arc is a set of \(k\) points such that \(k\geq n+1\) but not \(n+1\) points belong to a hyperplane. A normal rational curve in \(\text{PG}(n,q)\), \(2\leq n\leq q-2\) is a \(q+1\) arc projectively equivalent to the set of points \(\{(1,t, \cdots, t^ n)\mid t\in \text{GF} (q)^ +\}\) where \(\text{GF}(q)^ +\) denotes \(\text{GF}(q)\cup \{\infty\}\). Some of the main results: Theorem 2.1. A \(k\)-arc in \(\text{PG}(n,q)\), \(k\geq n+4\) and a dual \(k\)- arc \(\widehat{K}\) in \(\text{PG} (k-n-2,q)\) have isomorphic collineation groups. Theorem 3.2. In \(\text{PG}(q-2,q)\), \(q\) even, any point \(p\) which extends a normal rational curve \(K\) to a \(q+2\)-arc belongs to all osculating hyperplanes of \(K\).