$$\mathbb C^{2n-1}$$ 中 n 维完全相交的对称缺陷

L. R. G. Dias, Z. Jelonek
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摘要

让 \(X, Y \subset \mathbb {C}^{2n-1}\) 是一般位置上的 n 维强完全相交。在本注中,我们考虑连接点 \(x \in X) 和点 \(y \in Y) 的弦的中点集。这个集合被定义为映射的映像(Phi (x,y)=\frac{x+y}{2}.在 X 和 Y 的几何条件下,我们证明 X 和 Y 的对称缺陷,即映射 \(\Phi \) 的分叉集 B(X, Y) 是一个代数簇,其特征是拓扑不变式。我们引入了一个近似集 B(X,Y)的超曲面,并给出了它的度数估计。此外,对于任意两个 n 维的强完全相交 \(X,Y\subset \mathbb {C}^{2n-1}\) (包括 \(X=Y\) 的情况),我们引入了 X 和 Y 的一般对称缺陷集 \(\tilde{B}(X,Y)\),它被定义为同构。集合 \(\tilde{B}(X,Y)\) 是一个代数簇。最后我们证明,在实数情况下,如果 X、Y 紧凑,那么集合 (\tilde{B}(X,Y)\)是一个超曲面,它只有 Thom-Boardman 奇点。特别是如果 X 是紧凑的,那么 \(\tilde{B}(X)\) 是一个超曲面,它只有 Thom-Boardman 奇点。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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Symmetry Defect of n- Dimensional Complete Intersections in $$\mathbb C^{2n-1}$$

Let \(X, Y \subset \mathbb {C}^{2n-1}\) be n-dimensional strong complete intersections in a general position. In this note, we consider the set of midpoints of chords connecting a point \(x \in X\) to a point \(y \in Y\). This set is defined as the image of the map \(\Phi (x,y)=\frac{x+y}{2}.\) Under geometric conditions on X and Y, we prove that the symmetry defect of X and Y, which is the bifurcation set B(XY) of the mapping \(\Phi \), is an algebraic variety, characterized by a topological invariant. We introduce a hypersurface that approximates the set B(XY) and we present an estimate for its degree. Moreover, for any two n-dimensional strong complete intersections \(X,Y\subset \mathbb {C}^{2n-1}\) (including the case \(X=Y\)) we introduce a generic symmetry defect set \(\tilde{B}(X,Y)\) of X and Y, which is defined up to homeomorphism. The set \(\tilde{B}(X,Y)\) is an algebraic variety. Finally we show that in the real case if XY are compact, then the set \(\tilde{B}(X,Y)\) is a hypersurface and it has only Thom-Boardman singularities. In particular if X is compact, then \(\tilde{B}(X)\) is a hypersurface, which has only Thom-Boardman singularities.

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