On trisecant lines to White surfaces

In this work we show that the only White surface in the projective 5-space having an excess of trisecant lines is the polygonal surface constructed by C. Segre. The proof follows the line of B.Gambier's beautiful approach to this question and is intended to give it modern rigour. This has some implications on the geometry of the generic point of the principal componant of the Hilbert scheme of 18 points in the projective plane special in degree 5.