A degree bound for the parametrization of a rational surface
Parameterizations of rational surfaces are studied here in order to find an upper bound for the smallest degree of such a parameterization in terms of the degree or the sectional genus of the surface. The main results are the following: Let \(F\) be a rational surface of degree \(d\); then for any parameterization \(P\) of \(F\) we have \[ \deg P \geq \sqrt{{d\over 2}}. \] There exist a parameterization \(P\) of \(F\) such that \(\deg P \leq 2(d^2-4d+1)^2+2d+2\). If the sectional genus of \(F\) is \(p_1=0\), then there exists a parameterization \(P\) of \(F\) with \(\deg P\leq d\), while when \(p_1>0\) there exists \(P\) with \(\deg P \leq 4{2p_1-\deg F-1\choose 2}+4p_1\). Moreover, we always have \((\deg P-1)(\deg P-2)\geq 2p_1\), and, if \(p_1\geq 2\), \(d\leq 4p_1+4\). The main tool in proving the above result is repeated adjunction on the surface.
scientific article; zbMATH DE number 1398624