Fujita’s very ampleness conjecture for singular toric varieties

We present a self-contained combinatorial approach to Fujita’s conjectures in the toric case. Our main new result is a generalization of Fujita’s very ampleness conjecture for toric varieties with arbitrary singularities. In an appendix, we use similar methods to give a new proof of an anologous toric generalization of Fujita’s freeness conjecture due to Fujino. Given an ample divisor D and any other Cartier divisor D on an algebraic variety, we can choose t sufficiently large so that tD + D is basepoint free or very ample. In either case, it is not easy say how large we must choose t in general. However, for D = KX , Fujita made the following conjectures. Fujita’s Conjectures Let X be an n-dimensional projective algebraic variety, smooth or with mild singularities, D an ample divisor on X. (i) For t ≥ n+ 1, tD +KX is basepoint free. (ii) For t ≥ n+ 2, tD +KX is very ample. The example X ∼= P, D ∼ H shows that Fujita’s conjectured bounds are best possible. For smooth varieties, the corresponding statements with “basepoint free” and “very ample” replaced by “nef” and “ample”, respectively, are consequences of Mori’s Cone Theorem [Fuj]. For divisors on smooth toric varieties, nefness and ampleness are equivalent to freeness and very ampleness, respectively, so Fujita’s conjectures follow immediately for smooth toric varieties. One can also deduce Fujita’s conjectures for smooth toric varieties by general (non-toric) cohomological arguments of Ein and Lazarsfeld in characteristic zero [EL], and Smith in positive characteristic [Sm1] [Sm2], again using the fact that ample divisors on smooth toric varieties are very ample. For toric varieties with arbitrary singularities, a strong generalization of Fujita’s freeness conjecture was proved by Fujino [Fu]. We follow the usual toric convention fixing KX = − ∑ Di, the sum of the T -invariant prime divisors each with coefficient -1, as a convenient representative of the canonical class. Fujino’s Theorem Let X be a projective n-dimensional toric variety not isomorphic to P. Let D,D be Q-Cartier divisors such that 0 ≥ D ≥ KX , D+D ′