On trisecant lines to white surfaces

Inspired by an argument of Gambier, we show that the only White surface of image having a 4-dimensional trisecant locus is the Segre polygonal surface. This allows us to deduce that the generic point of the principal component of the subvariety W18[5] of 18-tuples special in degree 5 of the Hilbert scheme of 18 points of the plane corresponds to a smooth 18-tuple of points in uniform position, not lying on any quartic. This refines, in this particular case, the general bound due to Coppo. We also give the number of trisecant lines, counted with multiplicity, which pass through the generic point of a White surface of non-Segre type.