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Abstract: We study Hilbert functions of certain non-reduced schemes A supported atfinite sets of points in projective space, in particular, fat point schemes. Wegive combinatorially defined upper and lower bounds for the Hilbert function ofA using nothing more than the multiplicities of the points and informationabout which subsets of the points are linearly dependent. When N=2, we givethese bounds explicitly and we give a sufficient criterion for the upper andlower bounds to be equal. When this criterion is satisfied, we give both asimple formula for the Hilbert function and combinatorially defined upper andlower bounds on the graded Betti numbers for the ideal defining A, generalizingresults of Geramita-Migliore-Sabourin 2006. We obtain the exact Hilbertfunctions and graded Betti numbers for many families of examples, interestingcombinatorially, geometrically, and algebraically. Our method works in anycharacteristic. AWK scripts implementing our results can be obtained atthis http URL .



Author: Susan Cooper, Brian Harbourne, Zach Teitler

Source: https://arxiv.org/



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