$$J$$-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.

MSC:

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

Keywords:

reducible surfaces; projectability
Full Text:

References:

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