On base loci of higher fundamental forms of toric varieties

[摘要] 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) -> P-n, sending t to the vector of Laurent monomials with exponents p(0), ..., p(n) is an element of Z(k). We prove that the m-th fundamental form of such an Xat a general point has non empty base locus if and only if the points p(i) lie on a suitable degree-maffine hypersurface. We then restrict to the case in which the points p(i) 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. (C) 2020 Elsevier B.V. All rights reserved.

[发布日期] 2020-12-01