The smooth surfaces in \({\mathbb{P}}^ 4\) without apparent triple points
The smooth surfaces in \({\mathbb{P}}^ 4\) without apparent triple points are the elliptic quintic scrolls and the surfaces contained in quadric hypersurfaces. The method to prove this theorem is to study the sectional genus of a surface. The author shows that a surface in \({\mathbb{P}}^ 4\) without apparent triple points and not of the types stated above has to be birationally ruled over a curve of genus 2 or more. Next, he shows that it is contained in a cubic hypersurface. Finally, by using a formula concerning quadrisecant lines he shows that such a surface without apparent triple points does not exist. The positive direction of the theorem is a by the way result of \textit{F. Severi}: ``Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a suo punti tripli apparenti'', Rend. Circ. Mat. Palermo, II. Ser. 15, 33-51 (1901).
surfaces contained in quadric hypersurfaces
scientific article; zbMATH DE number 4125571