Error terms for the motives of discriminant complements and a Cayley-Bacharach theorem

Abstract

In this paper we prove under some simplifying hypotheses questions of Picoco and Levinson-Ullery on Cayley-Bacharach sets. Our results imply that, under suitable hypotheses Cayley-Bacharach sets lie on curves of low degree. We then use these results to estimate error terms to the normalized motive of the space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$as $d$ grows to infinity. The error term can be expressed in terms of a certain `sum over points' on plane cubic curves and the associated Hodge structure can be expressed in terms of the cohomology of the moduli space of elliptic curves. We also prove convergence of the motive of degree $d$ hypersurfaces in $\mathbb{P}^n$ as $n$ grows to infinity as well as other results on discriminant complements of high dimensional varieties.