\(J\)-embeddable reducible surfaces. (English) Zbl 1253.14050

Let \(\mathbb P^n\) be the \(n\)-dimensional complex projective space. In this paper the authors want to give a complete classification of \(J\)-embeddable surfaces having at most two irreducible components. More precisely they prove (see Lemma 16 and Proposition 18) the following result: Let \(V\) be a non-degenerate surface in \(\mathbb P^n\), \(n\geq 5.\) Assume that for a generic \(4\)-dimensional linear subspace \(\Lambda\subset \mathbb P^n\) the linear projection \(\pi_{L}:\mathbb P^n\rightarrow \Lambda\) is such that \(\pi_{L|V}\) is a \(J\)-embedding of \(V\) , and that \(V\) has at most two irreducible components. Then \(V\) is in the following list: (1) \(V\) is a Veronese surface in \(\mathbb P^5\); (2) \(V\) is an irreducible cone; (3) \(V\) is the union of a Veronese surface in \(\mathbb P^5\) and a tangent plane to it; (4) \(V\) is the union of two cones having the same vertex ; (5) \(V\) is the union of a cone with vertex a point \(P\) and a plane passing though \(P\); (6) \(V\) is the union of an irreducible surface \(S\), such that the dimension of its linear span \(\langle S \rangle\) is \(4\) and \(S\) is contained in a \(3\)-dimensional cone having a line \(l\) as vertex, and a plane cutting \(\langle S \rangle\) along \(l\).
In Section 4, the reader can found some examples of \(J\)-embeddable surfaces and the proof of a result concerning Veronese surfaces which will be useful for the classification.
Section \(5\) concerns the cases in which \(\dim[A;B]\leq 4\), where \(A\) and \(B\) are irreducible surfaces, including the case \(A=B\). In particular, we find Lemma 16 and Proposition 18 that give the proof of the main theorem.
In section \(6\) we find some information about the classification of \(J\)-embeddable surfaces \(V =V_1\cup \dots V_r\), \(r\geq 3\), i.e., \(J\)-embeddable surfaces having at least three irreducible components. By Corollary \(4\) this property is equivalent to assume that \(\dim \mathrm{Sec}(V )\leq 4\). As any surface \(V\) is \(J\)-embeddable if \(\dim \langle V \rangle\geq 4\) they will assume that \(\dim \langle V \rangle \geq 5\). Note that \(V\) is \(J\)-embeddable if and only if \(\dim[V_i; V_j ]\leq 4\) for any \(i,j=1,\dots, r\) (see Corollary \(5\)).
However the classification consists in a long list of cases and subcases, so that the authors give some information about them.


14N05 Projective techniques in algebraic geometry
14J25 Special surfaces
14N20 Configurations and arrangements of linear subspaces
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