A new graphic characterization of non singular quadrics of PG\((r,q)\) (Q1841860)

A subset \(K\) of the \(r\)-dimensional projective space \(\text{PG}(r,q)\) of order \(q\) is said to be of class \([a,b,m,n]_1\), where \( 0 \leq a <b<m<n\leq q+1\) if the lines of \(\text{PG}(r,q)\) can meet \(K\) only in \(a,b,m\) or \(n\) points. Quadrics of \(\text{PG}(r,q)\), with \(r \geq 3\), except for elliptic quadrics when \(r\) is odd and singular elliptic quadrics when \(r\) is even, have been characterized by G. Tallini in two articles, which appeared in 1956, as \(k\)-sets of class \([0,1,2,q+1]_1\) having enough points. In this article, the authors present characterizations of quadrics under weaker conditions on the class. Firstly, consider a \(k\)-set in \(\text{PG}(2s,q)\), \(s \geq 2\), with \(k=(q^{2s}-1)/(q-1)\). If every point of \(K\) lies on exactly \(q^{2(s-1)}\) tangents and on at most \((q^{2s-2}-1)/(q-1)\) \(n\)-secant lines, then \(K\) is a non-singular quadric of \(\text{PG}(2s,q)\). Secondly, consider a \(k\)-set in \(\text{PG}(2s-1,q)\), \(s \geq 2\), with \(k=(q^{2s-1}-1)/(q-1)+q^{s-1}\). If each point of \(K\) lies on exactly \(q^{2s-3}-q^{s-2}\) tangents and on at most \((q^{2s-1}-1)/(q-1)+q^{s-2}\) \(n\)-secant lines, then \(K\) is a hyperbolic quadric of \(\text{PG}(2s-1,q)\).