An automorphism of a spherical building is called domestic if it maps no chamber to an opposite chamber. In previous work the classification of domestic automorphisms in large spherical buildings of types 𝖥_4, 𝖤_6, and 𝖤_7 have been obtained, and in the present paper we complete the classification of domestic automorphisms of large spherical buildings of exceptional type of rank at least 3 by classifying such automorphisms in the 𝖤_8 case. Applications of this classification are provided, including Density Theorems showing that each conjugacy class in a group acting strongly transitively on a spherical building intersects a very small number of B-cosets, with B the stabiliser of a fixed choice of chamber.
For q = p^n with p an odd prime, the projective linear group PGL(2,q) can be seen as the stabilizer of a conic O in a projective plane π= PG(2,q). In that setting, involutions of PGL(2,q) correspond bijectively to points of π not in O. Triples of involutions { α_P,α_Q,α_R } of PGL(2,q) can then be seen also as triples of points {P,Q,R} of π. We investigate the interplay between algebraic properties of the group H = ⟨ α_P,α_Q,α_R ⟩ generated by three involutions and geometric properties of the triple of points {P,Q,R}. In particular, we show that the coset geometry Γ= Γ(H,(H_0,H_1,H_2)), where H_0 = ⟨ α_Q,α_R ⟩, H_1 = ⟨ α_P,α_R ⟩ and H_2 = ⟨ α_P,α_Q ⟩ is a regular hypertope if and only if {P,Q,R} is a strongly non self-polar triangle, a terminology we introduce. This entirely characterizes hypertopes of rank 3 with automorphism group a subgroup of PGL(2,q). As a corollary, we obtain the existence of hypertopes of rank 3 with non linear diagrams and with automorphism group PGL(2,q), for any q = p^n with p an odd prime. We also study in more details the case where the triangle {P,Q,R} is tangent to O.
In this paper we focus on various aspects of singular complex plane curves, mostly in the context of their homological properties and the associated combinatorial structures. We formulate some challenging open problems that can point to new directions in research, for example by introducing weak Ziegler pairs of curve arrangements. Moreover, we construct new examples of different Ziegler pairs, in both the classical and the weak sense, and present new geometric approaches to construction problems of singular plane curves.
Let A be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction A” of A to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. We initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of reflection arrangements. More precisely, let A = A(W) be the reflection arrangement of a complex reflection group W. By work of Terao, each such reflection arrangement is free. Thus so is Ziegler's canonical multiplicity on the restriction A” of A to a hyperplane. We show that the latter is inductively free as a multiarrangement if and only if A” itself is inductively free.
