Arrangements of codimension-one submanifolds

Face enumeration for line arrangements in a 2-torus

“…From left: toric arrangements with f -vectors (4,8,4), (4,9,5) and (4,10,6) Figure 6. From left: toric arrangements with f -vectors (4,11,7) and (4,12,8) Combining above inequality with Theorem 2.7 we see that for an arrangement of n toric lines f 0 ∈ { n−1 2 } ∪ {l : l ≥ n − 2}. As f 0 is not bounded above there is no hope for a complete characterization of the pairs (n, f 0 ).…”

“…Recently, Shnurnikov has characterized the set of all possible values of f 2 for toric line arrangements in [11]. See also [12] for arrangements in hyperbolic spaces, icosahedron and also arrangements of immersed circles in surfaces.…”

Manifold arrangements

“…The minimal and maximal number of connected components of arrangements has been studied by Shnurnikov [30] in the cases of Euclidean, projective and Lobachevskiȋ spaces. Moci [27] introduced a generalized Tutte polynomial for spherical and toric arrangements.…”

On the number of domains of maximal dimension in partitions of projective spaces by hyperplane arrangements

“…(The first one of them naturally coincides with the Shannon number.) In contrast to Martinov [2] and Shnurnikov [8], here we consider hyperplane arrangements in RP d for d ≥ 3. For d = 3, the first three numbers in F d (n) given by the theorem do not exceed 6n − 16 and coincide with the numbers found in [8].…”