Some finiteness results on Oda's question for pairs of line bundles

Abstract For a projective toric variety and nef line bundles on it, we prove some finiteness results related to Oda's question, which concerns the surjectivity of the multiplication map . When is general in the sense that there are no parallel edges on the polytopes associated to ample line bundles on it, we show that can be bounded by some fixed number independent of and . When is a smooth projective toric threefold, we prove that there exists a finite set of ample line bundles such that is surjective for any ample line bundle not in this set and any nef line bundle . In the proofs, we make use of coverings and quasi‐coverings of polytopes together with a filtered structure on that is gained from the restrictions of nef line bundles to the invariant curves on .