Algebraic characteristic sets of matroids

Abstract For a matroid M , define the algebraic characteristic set χA ( 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 χA ( PG (2, p )) and χA ( 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 .