  We present two algorithms determining all the complete and simplicial fans
admitting a fixed non-degenerate set of vectors $V$ as generators of their
1-skeleton. The interplay of the two algorithms allows us to discerning if the
associated toric varieties admit a projective embedding, in principle for any
values of dimension and Picard number. The first algorithm is slower than the
second one, but it computes all complete and simplicial fans supported by $V$
and lead us to formulate a topological-combinatoric conjecture about the
definition of a fan.
  On the other hand, we adapt the Sturmfels' arguments on the Gr\"obner fan of
toric ideals to our complete case; we give a characterization of the Gr\"obner
region and show an explicit correspondence between Gr\"obner cones and chambers
of the secondary fan. A homogenization procedure of the toric ideal associated
to $V$ allows us to employing GFAN and related software in producing our second
algorithm. The latter turns out to be much faster than the former, although it
can compute only the projective fans supported by $V$. We provide examples and
a list of open problems. In particular we give examples of rationally
parametrized families of $\Q$-factorial complete toric varieties behaving in
opposite way with respect to the dimensional jump of the nef cone over a
special fibre.
