Rational normal octavic surfaces with a double line, in space of five dimensions

Abstract

The following paper arises from a remark in a recent paper by Professor Baker. In that paper he gives a simple rule, under which a rational surface has a multiple line, expressed in terms of the system of plane curves which represent the prime sections of the surface. The rule is that, if one system of representing curves is given by an equation of the formthe surface being given, in space (x0, x1,…, xr), by the equationsthen the surface contains the linecorresponding to the curve φ = 0; and if the curve φ = 0 has genus q, this line is of multiplicity q + 1.