{"title":"$$\\mathbb C^{2n-1}$$ 中 n 维完全相交的对称缺陷","authors":"L. R. G. Dias, Z. Jelonek","doi":"10.1007/s00574-024-00414-7","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(X, Y \\subset \\mathbb {C}^{2n-1}\\)</span> be <i>n</i>-dimensional strong complete intersections in a general position. In this note, we consider the set of midpoints of chords connecting a point <span>\\(x \\in X\\)</span> to a point <span>\\(y \\in Y\\)</span>. This set is defined as the image of the map <span>\\(\\Phi (x,y)=\\frac{x+y}{2}.\\)</span> Under geometric conditions on <i>X</i> and <i>Y</i>, we prove that the symmetry defect of <i>X</i> and <i>Y</i>, which is the bifurcation set <i>B</i>(<i>X</i>, <i>Y</i>) of the mapping <span>\\(\\Phi \\)</span>, is an algebraic variety, characterized by a topological invariant. We introduce a hypersurface that approximates the set <i>B</i>(<i>X</i>, <i>Y</i>) and we present an estimate for its degree. Moreover, for any two <i>n</i>-dimensional strong complete intersections <span>\\(X,Y\\subset \\mathbb {C}^{2n-1}\\)</span> (including the case <span>\\(X=Y\\)</span>) we introduce a generic symmetry defect set <span>\\(\\tilde{B}(X,Y)\\)</span> of <i>X</i> and <i>Y</i>, which is defined up to homeomorphism. The set <span>\\(\\tilde{B}(X,Y)\\)</span> is an algebraic variety. Finally we show that in the real case if <i>X</i>, <i>Y</i> are compact, then the set <span>\\(\\tilde{B}(X,Y)\\)</span> is a hypersurface and it has only Thom-Boardman singularities. In particular if <i>X</i> is compact, then <span>\\(\\tilde{B}(X)\\)</span> is a hypersurface, which has only Thom-Boardman singularities.</p>","PeriodicalId":501417,"journal":{"name":"Bulletin of the Brazilian Mathematical Society, New Series","volume":"20 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Symmetry Defect of n- Dimensional Complete Intersections in $$\\\\mathbb C^{2n-1}$$\",\"authors\":\"L. R. G. Dias, Z. Jelonek\",\"doi\":\"10.1007/s00574-024-00414-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span>\\\\(X, Y \\\\subset \\\\mathbb {C}^{2n-1}\\\\)</span> be <i>n</i>-dimensional strong complete intersections in a general position. In this note, we consider the set of midpoints of chords connecting a point <span>\\\\(x \\\\in X\\\\)</span> to a point <span>\\\\(y \\\\in Y\\\\)</span>. This set is defined as the image of the map <span>\\\\(\\\\Phi (x,y)=\\\\frac{x+y}{2}.\\\\)</span> Under geometric conditions on <i>X</i> and <i>Y</i>, we prove that the symmetry defect of <i>X</i> and <i>Y</i>, which is the bifurcation set <i>B</i>(<i>X</i>, <i>Y</i>) of the mapping <span>\\\\(\\\\Phi \\\\)</span>, is an algebraic variety, characterized by a topological invariant. We introduce a hypersurface that approximates the set <i>B</i>(<i>X</i>, <i>Y</i>) and we present an estimate for its degree. Moreover, for any two <i>n</i>-dimensional strong complete intersections <span>\\\\(X,Y\\\\subset \\\\mathbb {C}^{2n-1}\\\\)</span> (including the case <span>\\\\(X=Y\\\\)</span>) we introduce a generic symmetry defect set <span>\\\\(\\\\tilde{B}(X,Y)\\\\)</span> of <i>X</i> and <i>Y</i>, which is defined up to homeomorphism. The set <span>\\\\(\\\\tilde{B}(X,Y)\\\\)</span> is an algebraic variety. Finally we show that in the real case if <i>X</i>, <i>Y</i> are compact, then the set <span>\\\\(\\\\tilde{B}(X,Y)\\\\)</span> is a hypersurface and it has only Thom-Boardman singularities. In particular if <i>X</i> is compact, then <span>\\\\(\\\\tilde{B}(X)\\\\)</span> is a hypersurface, which has only Thom-Boardman singularities.</p>\",\"PeriodicalId\":501417,\"journal\":{\"name\":\"Bulletin of the Brazilian Mathematical Society, New Series\",\"volume\":\"20 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the Brazilian Mathematical Society, New Series\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00574-024-00414-7\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Brazilian Mathematical Society, New Series","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00574-024-00414-7","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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。
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(X, Y) of the mapping \(\Phi \), is an algebraic variety, characterized by a topological invariant. We introduce a hypersurface that approximates the set B(X, Y) 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 X, Y 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.