Abstract
We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud–Mustaţǎ–Stillman and Mustaţǎ on cohomology of toric and monomial ideals to obtain a formula for computing χ(OX(D)) for a Weil divisor D on a complete simplicial toric variety XΣ. The main point is to use Alexander duality to pass from the toric irrelevant ideal, which appears in the computation of χ(OX(D)), to the Stanley–Reisner ideal of Σ, which is used in defining the Chow ring of XΣ.
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