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Pillow degenerations of K3 surfaces. (English) Zbl 1006.14014

Ciliberto, Ciro (ed.) et al., Applications of algebraic geometry to coding theory, physics and computation. Proceedings of the NATO advanced research workshop, Eilat, Israel, February 25-March 1, 2001. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys. Chem. 36, 53-63 (2001).
From the introduction: We construct a specific projective degeneration of K3 surfaces of degree \(2g-2\) in \(\mathbb{P}^g\) to a union of \(2g-2\) planes, which meet in such a way that the combinatorics of the configuration of planes is a triangulation of the 2-sphere. Abstractly, such degenerations are said to be type III degenerations of K3 surfaces. Although the birational geometry of such degenerations is fairly well understood, the study of projective degenerations is not nearly as completely developed.
C. Ciliberto, A. Lopez and R. Miranda [Invent. Math. 114, No. 3, 641-667 (1993; Zbl 0807.14028)] constructed projective degenerations of K3 surfaces to unions of planes in which the general member was embedded by a primitive line bundle. In this article we construct degenerations for which the general member is embedded by a multiple of the primitive line bundle class. The specific degenerations which we construct can be viewed as two rectangular arrays of planes, joined along their boundary, for this reason we have given them the name “pillow” degenerations. They are described in section 3. Following that, in section 4, we study the degeneration of the general branch curve (for a general projection of the surfaces to a plane) to a union of lines (which is the “branch curve” for the union of planes). In particular when the general branch curve is a plane curve having only nodes and cusps as singularities, we describe the degeneration of the nodes and the cusps to the configuration of the union of lines.
For the entire collection see [Zbl 0971.00013].

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14D06 Fibrations, degenerations in algebraic geometry

Citations:

Zbl 0807.14028
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