On the non-uniqueness of [formula omitted]-cones of matroids

Abstract

Let M be a rank- r simple GF(q) -representable matroid. A q-cone of M is a matroid M′ that is constructed by embedding M in a hyperplane of PG(r,q) , adding a point p of PG(r,q) not on H, and then adding all the points of PG(r,q) that are on lines joining p to an element of M. If r(M)>2 and M is uniquely representable over GF(q) , then M′ is unique up to isomorphism. This note settles a question made explicit by Kung by showing that if r(M)=2 or if M is not uniquely representable over GF(q) , then M′ need not be unique.