J-embeddable reducible surfaces (enlarged version)

TL;DR

This paper classifies reducible algebraic surfaces in projective space that can be J-embedded, focusing on those with secant varieties of dimension at most 4, providing detailed results for two-component surfaces.

Contribution

It offers a detailed classification of J-embeddable reducible surfaces, especially for two components, and discusses the complexity for three or more components.

Findings

Classified reducible surfaces with secant variety dimension ≤ 4.

Provided detailed analysis for surfaces with two components.

Outlined the complexity for surfaces with three or more components.

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.