Self-dual projective toric varieties

Abstract

Let T be a torus over an algebraically closed field k of characteristic 0, and consider a projective T -module P(V ). We determine when a projective toric subvariety X ⊂ P(V ) is self-dual, in terms of the configuration of weights of V .

A simple combinatorial criterion for projective toric manifolds with dual defect

“…Let P 0 be the triangle with vertices {(0, 0, 0, 0), (2, 0, 0, 1), (1, 0, 0, 1)}, P 1 the triangle with vertices {(0, 0, 0, 0), (0, 2, 0, 1), (0, 1, 0, 1)}, P 2 the triangle with vertices {(0, 0, 0, 0), (0, 0, 2, 1), (0, 0, 1, 0)}, and let P be the 6-dimensional Cayley polytope P = P 0 * P 1 * P 2 . It follows from [7] that P has exactly 9 lattice points (which can be easily verified). The associated toric variety (X, L) is defective with dual defect equal to 1 (in fact, it is self-dual).…”

“…We end with an example derived from the results of [7] in the classification of selfdual toric varieties, and results of [9]. We construct a singular dual defect toric variety associated with a (Cayley) lattice polytope P of dimension n = 6 which does not have the structure of a Cayley polytope of lattice polytopes P 0 , .…”

Higher order selfdual toric varieties

“…, a N } ⊂ Z n and let X A ⊂ P N be the corresponding toric variety. The variety X A is an affine invariant of the configuration A by Proposition II.5.1.2 in [10] (see also Section 2 in [2]) and the dimension of X A equals the dimension of the affine span of A. Thus we can replace, without loss of generality, our configuration by the affinely isomorphic lattice configuration…”

“…If we add more than one point, then the surface cannot be 2-selfdual. Because if it were, then we would have c 2 2, but this is not possible since the surface is not 2-Cayley. However, if we add all three points, we get a 3-selfdual surface [19].…”

“…We denote by X A ⊂ P N the projective toric variety rationally parametrized by t → (t a 0 : · · · : t an ). We note that by Lemma 2.14 in [2], X A is non degenerate (i.e., not contained in any hyperplane) if and only if the points a i are distinct, and we will always assume this holds. A lattice configuration A is said to be complete if it consists of all the lattice points in its convex hull.…”