Summary: PETERSEN PLANE ARRANGEMENTS AND A SURFACE WITH
HIROTACHI ABO, HOLGER P. KLEY, AND CHRIS PETERSON
Abstract. We study configurations of 2-planes in P4 that are combinatorially
described by the Petersen graph. We discuss conditions for configurations to be
locally Cohen-Macaulay and describe the Hilbert scheme of such arrangements.
An analysis of the homogeneous ideals of these configurations leads, via linkage,
to a class of smooth, general type surfaces in P4. We compute their numerical
invariants and show that they have the unusual property that they admit
(multiple) 7-secants. Finally, we demonstrate that the construction applied
to Petersen arrangements with additional symmetry leads to surfaces with
exceptional automorphism groups.
Linkage (or liaison) theory can provide a bridge between combinatorially inter-
esting varieties such as -plane arrangements (equidimensional unions of projective
subspaces of Pn
) and geometrically interesting ones: smooth projective varieties.
This paper studies one such bridge, connecting the class of 2-plane arrangements
whose incidence structure may be described by the Petersen graph with a