The authors consider two notions of non-speciality for a smooth nondegenerate surface S in \({\mathbb{P}}^ 4:\)
The surface S is sectionally non-special if the general hyperplane section C is a non-special curve, i.e. \(h^ 1(C,{\mathcal O}_ C(1))=0;\)
The surface S is non-special if \(h^ 1(S,{\mathcal O}_ S(1))=0.\)
The authors prove that there is a finite number of such surfaces S and they give a complete list of them. - To obtain the first theorem on sectionally non-special surfaces they study the adjunction map \(\phi\), i.e. the map given by the linear system \(| K+H|\), where K is the canonical divisor on S and H is the hyperplane divisor, and they use some results of A. Sommese and A. Van de Ven on the map \(\phi\), and of P. Ellia and G. Sacchiero on the classification of the smooth conic bundles of \({\mathbb{P}}^ 4\). The authors consider the relations between the irregularity q of S, the sectional genus \(\pi\), i.e. the genus of the general hyperplane section C, and the dimension of the image \(\phi\) (S) of S by the adjunction map. - Then they can prove that the pair (d,\(\pi\)) can take only a finite number of values, where d is the degree of S and \(\pi\) the sectional genus of S. From this theorem the authors obtain also a description of the smooth varieties X of codimension 2 in \({\mathbb{P}}^ n\), \(n\geq 5\), such that the general intersection with a linear space of dimension \(3\) is a non-special curve.
To prove the theorem on non-special surfaces they assume that the surface S is not of general type, and by the preceding result they can suppose that the general hyperplane section C is special. - Then they use results on classification of surfaces.