Asymptotic Betti bounds for hypersurfaces in a singular variety

TL;DR

This paper establishes explicit asymptotic bounds on the Betti numbers of hypersurfaces within possibly singular projective varieties, improving known bounds and demonstrating sharpness in certain cases.

Contribution

It provides new explicit asymptotic bounds on Betti numbers of hypersurfaces in singular varieties, extending previous results and including bounds for constructible sheaves.

Findings

Betti number bounds for hypersurfaces in singular varieties

Improved bounds for local complete intersection cases

Asymptotic sharpness of the bounds

Abstract

We show that for any degree $d$ hypersurface $Y \subset X$ in a possibly singular projective variety $X \subset \mathbf{P}^N$, the total Betti number of $Y$ is bounded by $3\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$ for some explicit constant $C > 0$ independent of $d$ and $Y$. When $X$ is a local complete intersection, the bound improves to $\text{deg}(X)\cdot d^n + C\cdot d^{n-1}$. In this case, the bound is asymptotically sharp. Similar bounds are also established for general constructible sheaves.