Optimal upper bounds for anti-canonical volumes of singular toric Fano varieties

Fix two positive integers d≥3 and q. We give an upper bound for anti-canonical volumes of d-dimensional 1/q-lc toric Fano varieties, which corresponds to an upper bound for the dual normalized volumes of the associated d-dimensional 1/q-lc Fano polytopes. And we also construct examples to show that these upper bounds are optimal. Besides, we provide an optimal upper bound for volumes of d-dimensional lattice simplices S such that 1/qS has exactly one interior lattice point.