Existence of real algebraic hypersurfaces with many prescribed components

TL;DR

This paper demonstrates the existence of real algebraic hypersurfaces with many connected components and maximal Betti number growth, using probabilistic methods, and extends results to complete intersections.

Contribution

It constructs hypersurfaces with prescribed topology and maximal Betti numbers, employing probabilistic techniques, and generalizes to complete intersections.

Findings

Existence of hypersurfaces with many prescribed components

Maximal growth of Betti numbers in linear systems

Extension to complete intersections

Abstract

Given a real algebraic variety $X$ of dimension $n$, a very ample divisor $D$ on $X$ and a smooth closed hypersurface $\Sigma$ of $\mathbf{R}^n$, we construct real algebraic hypersurfaces in the linear system $|mD|$ whose real locus contains many connected components diffeomorphic to $\Sigma$. As a consequence, we show the existence of real algebraic hypersurfaces in the linear system $|mD|$ whose Betti numbers grow by the maximal order, as $m$ goes to infinity. As another application, we recover a result by D. Gayet on the existence of many disjoint lagrangians with prescribed topology in any smooth complex hypersurface of $\mathbf{C}\mathbf{P}^n$. The results in the paper are proved more generally for complete intersections. The proof of our main result uses probabilistic tools.