{"title":"真正的广义三十进制三分法","authors":"Kristian Ranestad, Anna Seigal, Kexin Wang","doi":"arxiv-2409.01356","DOIUrl":null,"url":null,"abstract":"The classical trisecant lemma says that a general chord of a non-degenerate\nspace curve is not a trisecant; that is, the chord only meets the curve in two\npoints. The generalized trisecant lemma extends the result to\nhigher-dimensional varieties. It states that the linear space spanned by\ngeneral points on a projective variety intersects the variety in exactly these\npoints, provided the dimension of the linear space is smaller than the\ncodimension of the variety and that the variety is irreducible, reduced, and\nnon-degenerate. We prove a real analogue of the generalized trisecant lemma,\nwhich takes the form of a trichotomy. Along the way, we characterize the\npossible numbers of real intersection points between a real projective variety\nand a complimentary dimension real linear space. We show that any integer of\ncorrect parity between a minimum and a maximum number can be achieved. We then\nspecialize to Segre-Veronese varieties, where our results apply to the\nidentifiability of independent component analysis, tensor decomposition and to\ntypical tensor ranks.","PeriodicalId":501063,"journal":{"name":"arXiv - MATH - Algebraic Geometry","volume":"50 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Real Generalized Trisecant Trichotomy\",\"authors\":\"Kristian Ranestad, Anna Seigal, Kexin Wang\",\"doi\":\"arxiv-2409.01356\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The classical trisecant lemma says that a general chord of a non-degenerate\\nspace curve is not a trisecant; that is, the chord only meets the curve in two\\npoints. The generalized trisecant lemma extends the result to\\nhigher-dimensional varieties. It states that the linear space spanned by\\ngeneral points on a projective variety intersects the variety in exactly these\\npoints, provided the dimension of the linear space is smaller than the\\ncodimension of the variety and that the variety is irreducible, reduced, and\\nnon-degenerate. We prove a real analogue of the generalized trisecant lemma,\\nwhich takes the form of a trichotomy. Along the way, we characterize the\\npossible numbers of real intersection points between a real projective variety\\nand a complimentary dimension real linear space. We show that any integer of\\ncorrect parity between a minimum and a maximum number can be achieved. We then\\nspecialize to Segre-Veronese varieties, where our results apply to the\\nidentifiability of independent component analysis, tensor decomposition and to\\ntypical tensor ranks.\",\"PeriodicalId\":501063,\"journal\":{\"name\":\"arXiv - MATH - Algebraic Geometry\",\"volume\":\"50 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Algebraic Geometry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.01356\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Algebraic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.01356","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The classical trisecant lemma says that a general chord of a non-degenerate
space curve is not a trisecant; that is, the chord only meets the curve in two
points. The generalized trisecant lemma extends the result to
higher-dimensional varieties. It states that the linear space spanned by
general points on a projective variety intersects the variety in exactly these
points, provided the dimension of the linear space is smaller than the
codimension of the variety and that the variety is irreducible, reduced, and
non-degenerate. We prove a real analogue of the generalized trisecant lemma,
which takes the form of a trichotomy. Along the way, we characterize the
possible numbers of real intersection points between a real projective variety
and a complimentary dimension real linear space. We show that any integer of
correct parity between a minimum and a maximum number can be achieved. We then
specialize to Segre-Veronese varieties, where our results apply to the
identifiability of independent component analysis, tensor decomposition and to
typical tensor ranks.