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arXiv · 1207.5467

Asymptotics of random Betti tables

Lawrence Ein·Daniel Erman·Robert Lazarsfeld
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Abstract

The purpose of this paper is twofold. First, we present a conjecture to the effect that the ranks of the syzygy modules of a smooth projective variety become normally distributed as the positivity of the embedding line bundle grows. Then, in an attempt to render the conjecture plausible, we prove a result suggesting that this is in any event the typical behavior from a probabilistic point of view. Specifically, we consider a "random" Betti table with a fixed number of rows, sampled according to a uniform choice of Boij-Soderberg coefficients. We compute the asymptotics of the entries as the length of the table goes to infinity, and show that they become normally distributed with high probability.

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Lawrence Ein, Daniel Erman, Robert Lazarsfeld. 2012-07-23. Asymptotics of random Betti tables. https://doi.org/10.1515/crelle-2013-0032

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