Toric varieties and Gröbner bases: the complete Q-factorial case

Abstract: 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.