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We present a self-contained combinatorial approach to Fujita's conjectures inthe toric case. Our main new result is a generalization of Fujita's veryampleness conjecture for toric varieties with arbitrary singularities. In anappendix, we use similar methods to give a new proof of an analogous toricgeneralization of Fujita's freeness conjecture due to Fujino.

Fujita's very ampleness conjecture for singular toric varieties