Asymptotics of random Betti tables

Abstract

AbstractThis paper is motivated by the question of understanding the asymptotic behavior of the Betti numbers of the resolution of the ideal of a projective variety as the positivity of the embedding line bundle grows. We present a conjecture asserting that these invariants approach a Gaussian distribution, and we verify this in the case of curves. Then we work out the asymptotics of “random” Betti tables with a fixed number of rows, sampled according to a uniform choice of Boij–Söderberg coefficients. This analysis suggests that the normal distribution of Betti numbers is in any event the typical behavior from a probabilistic viewpoint.