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

Alzati, Alberto; Ballico, Edoardo

Ark. Mat. 49, No. 2, 199-215 (2011).

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.

Reviewer: Elena Guardo (Catania)

MSC:

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

Keywords:

reducible surfaces; projectability

References:

[1]	Ådlandsvik, B., Joins and higher secant varieties, Math. Scand. 62 (1987), 213–222. · Zbl 0657.14034
[2]	Ådlandsvik, B., Varieties with an extremal number of degenerate higher secant varieties, J. Reine Angew. Math. 392 (1988), 16–26. · Zbl 0649.14029
[3]	Dale, M., Severi’s theorem on the Veronese-surface, J. London Math. Soc. 32 (1985), 419–425. · Zbl 0597.14030
[4]	Johnson, K. W., Immersion and embedding of projective varieties, Acta Math. 140 (1981), 49–74. · Zbl 0373.14005
[5]	Zak, F. L., Tangents and Secants of Algebraic Varieties, Translations of Mathematical Monographs 127, Amer. Math. Soc., Providence, RI, 1993. · Zbl 0795.14018