  We investigate possible linear, algebraic, and Frobenius flock characteristic
sets of matroids. In particular, we classify possible combinations of linear
and algebraic characteristic sets when the algebraic characteristic set is
finite or cofinite. We also show that the natural density of an algebraic
characteristic set in the set of primes may be arbitrarily close to any real
number in the interval $[0,1]$.
  Frobenius flock realizations can be constructed from algebraic realizations,
but the converse is not true. We show that the algebraic characteristic set may
be an arbitrary cofinite set even for matroids whose Frobenius flock
characteristic set is the set of all primes. In addition, we construct
Frobenius flock realizations in all positive characteristics from linear
realizations in characteristic 0, and also from Frobenius flock realizations of
the dual matroid.
