Combinatorial method in adjoint linear systems on toric varieties (Q1432724)

\textit{T. Fujita} conjectured [in: Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 167--178 (1987; Zbl 0659.14002)] that if \(D\) is an ample divisor on a nonsingular complex projective \(n\)-dimensional variety \(X\), then \(K_X+\ell D\) is globally generated for \(\ell\geq n+1\) and very ample for \(\ell\geq n+2\). For curves this is well known and for \(n = 2\) it follows from work by \textit{I. Reider} [Ann. Math. (2) 127, No. 2, 309--316 (1988; Zbl 0663.14010)]. ``Globally generated'' is known for \(n\leq 4\). For toric varieties the conjecture follows from Fujita's paper quoted above and work by \textit{K. Smith} [``Fujita's freeness conjecture in terms of local cohomology'', J. Algebr. Geom. 6, No. 3, 417--429 (1997; Zbl 0901.14005)]. In this paper the author gives a new treatment of the case of toric varieties, motivated by the philosophy that these results should admit elementary proofs using only toric combinatorial techniques. Similar results were obtained independently by \textit{M. Mustata} [Tohoku Math. J., II. Ser. 54, No. 3, 451--470 (2002; Zbl 1092.14064)].