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Patchworking real algebraic hypersurfaces with asymptotically large Betti numbers
Journal of Topology ( IF 1.1 ) Pub Date : 2022-06-23 , DOI: 10.1112/topo.12251
Charles Arnal 1
Affiliation  

In this article, we describe a recursive method for constructing a family of real projective algebraic hypersurfaces in ambient dimension n $n$ n from families of such hypersurfaces in ambient dimensions k = 1 , , n 1 $k=1,\ldots ,n-1$ k=1,,n1. The asymptotic Betti numbers of real parts of the resulting family can then be described in terms of the asymptotic Betti numbers of the real parts of the families used as ingredients. The algorithm is based on Viro's Patchwork (Patchworking real algebraic varieties, 2006) and inspired by Itenberg's and Viro's construction of asymptotically maximal families in arbitrary dimension (Proceedings of Gökova Geometry-Topology Conference, 2006). Using it, we prove that for any n $n$ n and i = 0 , , n 1 $i=0,\ldots ,n-1$ i=0,,n1, there is a family of asymptotically maximal real projective algebraic hypersurfaces { Y d n } d $\lbrace Y^n_d\rbrace _d$ {Ynd}d in R P n ${\mathbb {R}}{\mathbb {P}}^n$ (where d $d$ denotes the degree of Y d n $Y^n_d$ ) such that the i $i$ th Betti numbers b i ( R Y d n ) $b_i({\mathbb {R}}Y^n_d)$ are asymptotically strictly greater than the ( i , n 1 i ) $(i,n-1-i)$ th Hodge numbers h i , n 1 i ( C Y d n ) $h^{i,n-1-i}({\mathbb {C}}Y^n _d)$ . We also build families of real projective algebraic hypersurfaces whose real parts have asymptotic (in the degree d $d$ ) Betti numbers that are asymptotically (in the ambient dimension n $n$ ) very large.

中文翻译:

具有渐近大 Betti 数的拼凑实代数超曲面

在本文中,我们描述了一种在环境维度上构造实数投影代数超曲面族的递归方法 n $n$ n来自环境维度中此类超曲面的族 ķ = 1 , , n - 1 $k=1,\ldots ,n-1$ k = 1 , ... , n - 1. 然后可以根据用作成分的族的实部的渐近 Betti 数来描述所得族的实部的渐近 Betti 数。该算法基于 Viro 的 Patchwork(Patchworking 实数代数变体,2006 年),并受到 Itenberg 和 Viro 在任意维度上构建渐近最大族的启发(Gökova几何拓扑会议论文集,2006 年)。使用它,我们证明对于任何 n $n$ n 一世 = 0 , , n - 1 $i=0,\ldots ,n-1$ = 0 , ... , n - 1, 有一族渐近极大实射影代数超曲面 { d n } d $\lbrace Y^n_d\rbrace _d$ {nd}d R n ${\mathbb {R}}{\mathbb {P}}^n$ (在哪里 d $d$ 表示程度 d n $Y^n_d$ ) 使得 一世 $i$ 贝蒂号码 b 一世 ( R d n ) $b_i({\mathbb {R}}Y^n_d)$ 渐近严格地大于 ( 一世 , n - 1 - 一世 ) $(i,n-1-i)$ 霍奇数 H 一世 , n - 1 - 一世 ( C d n ) $h^{i,n-1-i}({\mathbb {C}}Y^n _d)$ . 我们还建立了实数投影代数超曲面族,其实部具有渐近线(在度 d $d$ ) 渐近的 Betti 数(在环境维度中) n $n$ ) 很大。
更新日期:2022-06-27
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