Study on Minkowski sums of lattice polytoped and toric varieties

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The aim of this research is to prove the conjecture that all ample line bundles on a nonsingular toric variety in dimension three and four are normally generated.In 2011, our article which proves that an ample line bundle on a nonsingular toric 3-fold whose adjoint bundle is not big is normally generated has been printed in a scientific journal. We construct examples of very ample but not normal lattice polytopes in all dimensions greater than two. And we proves that all ample line bundles on a nonsingular toric 3-fold which has a surjective morphism onto the projective line are normally generated. In 2013, we proves that all ample lines bundles on a polarized toric 3-fold which has an ample line bundle with non-big adjoint bundle are normally generated.