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Algebro-geometric stability and crystals. (English) Zbl 1292.52024

Summary: In this survey, the author reports the recent applications in discrete geometric analysis of his joint work with Seshadri in the 1970’s on the compactifications of the generalized Jacobian variety of a nodal curve by means of Geometric Invariant Theory. The algebro-geometric stability is described in terms of the dual graph of the nodal curve. The description leads to convex polyhedral facet-to-facet and space-filling tilings, called Namikawa tilings, of Euclidean spaces with respect to lattices and to the combinatorial description of the compactifications.
The convex polyhedral facet-to-facet and space-filling Namikawa tilings generalizing those above and appearing in the more general orthonormal setting, not necessarily coming from graphs, turn out to be related to the “cut and project” method for crystals and quasicrystals.
The crystals constructed as the standard realization of finite graphs by Kotani and Sunada turn out to have hidden in it a Voronoi tiling and a Namikawa tiling at least in the case of maximal abelian coverings of the graph. Guided by examples, the author poses a question and formulates a conjecture in the case of non-maximal free abelian coverings.

MSC:

52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
05C40 Connectivity
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14T05 Tropical geometry (MSC2010)
52C23 Quasicrystals and aperiodic tilings in discrete geometry
74E15 Crystalline structure
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82D25 Statistical mechanics of crystals
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