Curves of the projective 3-space, tangent developables and partial spreads (Q1590256)

The authors are interested here in curves \(X\) of \(P(3,K)\) where \(K\) is an algebraic closed field of characteristic \(p \geq 0\) satisfying the following condition: (*) Tangent lines to \(X\) at distinct points are skew. In the case when \(K\) is the algebraic closure of \(GF(q)\), the above condition means that the tangent lines to \(X\) at \(GF(q)\)-rational points will form a (partial) spread of \(PG(3,q)\). Under suitable assumptions the authors prove that if the curve \(X\) satisfies the condition (*) then \(X\) must necessarily be a twisted cubic and they also find an infinite family of curves of \(PG(3,K)\) distinct from the twisted cubic, satisfying property (*). In order to do that, they characterize twisted cubic. More precisely, for a smooth degree \(d\) curve \(X\) such that for a general point \(P \in X\) there is no tangent line to \(X\) at a point \(Q\neq P\), with \(T_PX \cap T_QX \neq \empty\), they prove that \(X\equiv PG(1,K)\), and either \(d=3\) and \(X\) is a twisted cubic, or \(d=p^e+1\) and \(x\) is projectively equivalent to the rational curve \(D\) with the parametrization \[ (w_0,w_1)\mapsto (w_0^{p^e+1},w_0^{p^e}w_1,w_0w_1^{p^e},w_1^{p^e+1}). \] The authors also prove that two tangent lines to \(X\), where \(X\) is one of the above curves, are skew (if \(X\) is the curve \(D\), they also require \(p^e\geq 4\)).