On base loci of higher fundamental forms of toric varieties

We study the base locus of the higher fundamental forms of a projective toric variety X at a general point. More precisely we consider the closure X of the image of a map (C*)k→Pn, sending t to the vector of Laurent monomials with exponents p0,…,pn∈Zk. We prove that the m-th fundamental form of such an X at a general point has non empty base locus if and only if the points pi lie on a suitable degree-m affine hypersurface. We then restrict to the case in which the points pi are all the lattice points of a lattice polytope and we give some applications of the above result. In particular we provide a classification for the second fundamental forms on toric surfaces, and we also give some new examples of weighted 3-dimensional projective spaces whose blowing up at a general point is not Mori dream.