J-embeddable reducible surfaces (enlarged version)

Abstract

Let V be a variety in P^n(C) and let W be a linear space, of dimension w, in P^n. We say that V can be isomorphically projected onto W if there exists a linear projection f, from a suitable linear space L disjoint from W, dim(L) = n-w-1 >= 0, such that f(V) is isomorphic to V. Let f' be the restriction of f to V. We say that f' is a J-embedding of V (see K. W. Johnson: Immersion and embedding of projective varieties, Acta Math. [140] (1981) 49-74) if f' is injective and the differential of f' is a finite map. In this paper we classify the J-embeddable reducible surfaces, equivalently, the reducible surfaces whose secant variety has dimension at most 4. The classification is very detailed for surfaces having two components. For three or more components we give a reasonable classification, taking into account that the complete classification is very rich of cases, subcases and subsubcases.