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On complete degenerations of surfaces with ordinary singularities in \( \mathbb P^3\). (English. Russian original) Zbl 1205.14011
Sb. Math. 201, No. 1, 129-158 (2010); translation from Mat. Sb. 201, No. 1, 129-160 (2010).
The article targets a degeneration problem for projective surfaces of the 3-dimensional projective space. The model of the investigation is the lower dimensional case, namely, Severi’s theorem (proved by J. Harris in [Invent. Math. 84, 445–461 (1986; Zbl 0596.14017)]). The statement is that any nodal projective plane curves of the projective plane can be deformed into an arrangement of lines in general position. Such a deformation defines a set of limit double points, as the limit of the nodes of the family of nodal curves. Conversely, G.-M. Greuel, Ch. Lossen and E. Shustin [Introduction to singularities and deformations. Springer Monographs in Mathematics. Berlin: Springer (2007; Zbl 1125.32013)] proved that any subset of the nodes of an arrangement might appear as a limit of nodes of some flat deformation (that is, the other nodes can be smoothed).
The authors investigate the analogue of the above two statements in one dimension higher. More precisely, they consider a surface in the three-dimensional projective space with only ordinary singularities (that is, only with a double curve, triple and pinch points). These are the singularities which appear on the generic projection of a surface into a three-dimensional space. Then they ask the possibility to degenerate such a surface into an arrangement such that the double curve degenerates flatly into a subset of the double curve of the arrangement, and the triple point also behave similarly. The output of the article is that the analogues of the two statements, valid for curves, are obstructed in the case of surfaces. In order to show this, the authors study several numerical characteristics of surfaces with ordinary singularities and their degenerations.

MSC:
14D06 Fibrations, degenerations in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
32S22 Relations with arrangements of hyperplanes
32S30 Deformations of complex singularities; vanishing cycles
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