Algebraic characteristic sets of matroids

For a matroid M, define the algebraic characteristic set x4(M) to be the set of field characteristics over which M can be algebraically represented. We construct many examples of rank three matroids with finite, non-singleton algebraic characteristic sets. We also determine x,,(PG(Z, p)) and xa(AG(2, p)). An infinite family of rank three matroids with empty algebraic characteristic set is constructed. In addition, we answer some antichain and excluded minor questions for algebraic representability over a given field F.