Cohomology on Toric Varieties and Local Cohomology with Monomial Supports

2000/01/27 by David Eisenbud, Mircea Mustata, Mike Stillman · 1 citation
Mathematics · #math.AG #math.AC #msc:14M25 #msc:13D45 #msc:14Q99

paper · pdf

published as J. Symbolic Comput. 29 (2000), no. 4-5, 583-600 · 23 pages, 2 figures, uses diagrams.tex, to appear in Journal of Symbolic Computation

arxiv created 2000/01/27 · arxiv updated 2009/11/30

Abstract

We study the local cohomology modules Hⁱ_B(R) for a reduced monomial ideal B in a polynomial ring R=k[X₁,...,X_n]. We consider a grading on R which is coarser than the Zⁿ-grading such that each component of Hⁱ_B(R) is finite dimensional and we give an effective way to compute these components. Using Cox's description for sheaves on toric varieties, we apply these results to compute the cohomology of coherent sheaves on toric varieties. We give algorithms for this computation which have been implemented in the Macaulay 2 system. We obtain also a topological description for the cohomology of rank one torsionfree sheaves on toric varieties.

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