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Asymptotic syzygies of normal crossing varieties

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      Abstract

      Asymptotic syzygies of a normal crossing variety follow the same vanishing behavior as one of its smooth components, unless there is a cohomological obstruction arising from how the smooth components intersect each other. In that case, we compute the asymptotic syzigies in terms of the cohomology of the simplicial complex associated to the normal crossing variety.

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      Most cited references 5

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      Koszul cohomology and the geometry of projective varieties

       Mark L. Green (1984)
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        Asymptotic syzygies of algebraic varieties

        This paper studies the asymptotic behavior of the syzygies of a smooth projective variety X as the positivity of the embedding line bundle grows. We prove that as least as far as grading is concerned, the minimal resolution of the ideal of X has a surprisingly uniform asymptotic shape: roughly speaking, generators eventually appear in almost all degrees permitted by Castelnuovo-Mumford regularity. This suggests in particular that a widely-accepted intuition derived from the case of curves -- namely that syzygies become simpler as the degree of the embedding increases -- may have been misleading. For Veronese embeddings of projective space, we give an effective statement that in some cases is optimal, and conjecturally always is so. Finally, we propose a number of questions and open problems concerning asymptotic syzygies of higher-dimensional varieties.
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          On the cohomology of a simple normal crossings divisor

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            Author and article information

            Journal
            2017-01-23
            1701.06275

            http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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            math.AG

            Geometry & Topology

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