Petersen Plane Arrangements and a Surface with Multiple 7-secants

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.