论文标题
拼凑的真实代数超曲面具有渐近较大的betti数字
Patchworking real algebraic hypersurfaces with asymptotically large Betti numbers
论文作者
论文摘要
在本文中，我们介绍了一种递归方法，用于在环境尺寸的环境中$ n $中构建一个真实的投射代数超曲面$ k = 1，\ ldots，n-1 $。然后可以用用作成分的家族的实际部分的渐近betti数量来描述所得家族的实际部分的渐近betti数量。 该算法基于Viro的拼布，并受到I. Itenberg和O. viro的启发。 使用它，我们证明，对于任何$ n $和$ i = 0，\ ldots，n-1 $，都有一个渐近最大的实际投影型代数hypersurfaces $ \ {y^n_d \} _d $ $ i $ -th betti数字$ b_i（\ mathbb {r} y^n_d）$渐近地比$（i，n-1-i）$ - hodge数字$ h^{i，n-1-i}（\ mathbb {c} y^n_d）$。我们还建立了真正的投射代数超曲面的家族，其实际零件具有渐近的betti数字，这些数字是渐近的（在环境尺寸$ n $中）。
In this article, we describe a recursive method for constructing a family of real projective algebraic hypersurfaces in ambient dimension $n$ from families of such hypersurfaces in ambient dimensions $k=1,\ldots,n-1$. 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 and inspired by I. Itenberg's and O. Viro's construction of asymptotically maximal families in arbitrary dimension. Using it, we prove that for any $n$ and $i=0,\ldots,n-1$, there is a family of asymptotically maximal real projective algebraic hypersurfaces $\{Y^n_d\}_d$ in $\mathbb{R} \mathbb{P} ^n$ (where $d$ denotes the degree of $Y^n_d$) such that the $i$-th Betti numbers $b_i(\mathbb{R} Y^n_d)$ are asymptotically strictly greater than the $(i,n-1-i)$-th Hodge numbers $h^{i,n-1-i}(\mathbb{C} Y^n_d)$. We also build families of real projective algebraic hypersurfaces whose real parts have asymptotic (in the degree $d$) Betti numbers that are asymptotically (in the ambient dimension $n$) very large.
