naa a22 4500 510736629 CHVBK 20180411083012.0 cr unu---uuuuu 180411e20130301xx s 000 0 eng 10.1007/s13366-011-0064-4 doi (NATIONALLICENCE)springer-10.1007/s13366-011-0064-4 Graph varieties in high dimension [Elektronische Daten] [Thomas Enkosky, Jeremy Martin] We study the picture space $${\mathcal{X}^d(G)}$$ of all embeddings of a finite graph G as point-and-line arrangements in an arbitrary-dimensional projective space, continuing previous work on the planar case. The picture space admits a natural decomposition into smooth quasiprojective subvarieties called cellules, indexed by partitions of the vertex set of G, and the irreducible components of $${\mathcal{X}^d(G)}$$ correspond to cellules that are maximal with respect to a partial order on partitions that is in general weaker than refinement. We study both general properties of this partial order and its characterization for specific graphs. Our results include complete combinatorial descriptions of the irreducible components of the picture spaces of complete graphs and complete multipartite graphs, for any ambient dimension d. In addition, we give two graph-theoretic formulas for the minimum ambient dimension in which the directions of edges in an embedding of G are mutually constrained. The Managing Editors, 2011 Graph variety nationallicence Picture space nationallicence Set partition nationallicence Ear decomposition nationallicence Enkosky Thomas Department of Mathematics, University of Kansas, 66047, Lawrence, KS, USA aut Martin Jeremy Department of Mathematics, University of Kansas, 66047, Lawrence, KS, USA aut Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry Springer-Verlag 54/1(2013-03-01), 1-12 0138-4821 54:1<1 2013 54 13366 https://doi.org/10.1007/s13366-011-0064-4 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s13366-011-0064-4 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Enkosky Thomas Department of Mathematics, University of Kansas, 66047, Lawrence, KS, USA aut NATIONALLICENCE 700 1- Martin Jeremy Department of Mathematics, University of Kansas, 66047, Lawrence, KS, USA aut NATIONALLICENCE 773 0- Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry Springer-Verlag 54/1(2013-03-01), 1-12 0138-4821 54:1<1 2013 54 13366 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer