Asymptotic Betti bounds for hypersurfaces in a singular variety
Xuanyu Pan, Dingxin Zhang, Xiping Zhang
TL;DR
This work establishes uniform asymptotic upper bounds for the total Betti numbers of degree d hypersurfaces in arbitrary projective varieties X, extending classical bounds from projective space to singular ambient spaces. The authors develop a perverse-sheaf framework and introduce the Proper Degeneration Lemma to control cohomology across all degrees, yielding bounds of the form B(Y, F|_Y) ≲ deg(X) · d^n with explicit lower order corrections, and with improved constants when X is a local complete intersection. They further show that these bounds hold for general constructible ℓ-adic sheaves and achieve uniformity in the coefficient ℓ, under compatible systems and finite-field settings. The results connect perverse-weak-Lefschetz theory, characteristic classes, and degeneration techniques to provide sharp, scalable estimates with applications to arithmetic geometry and equidistribution problems. Overall, the paper extends Betti-number control from smooth projective spaces to broad singular settings and clarifies when the constants can be made independent of the ℓ-adic coefficient field."
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.