Pub Date : 2017-01-01Epub Date: 2017-02-07DOI: 10.1007/s12220-017-9762-y
Ivan Cheltsov
We prove that and are the smallest log canonical thresholds of reduced plane curves of degree , and we describe reduced plane curves of degree d whose log canonical thresholds are these numbers. As an application, we prove that and are the smallest values of the -invariant of Tian of smooth surfaces in of degree . We also prove that every reduced plane curve of degree whose log canonical threshold is smaller than is GIT-unstable for the action of the group , and we describe GIT-semistable reduced plane curves with log canonical thresholds .
我们证明2 d, 2 d - 3 (d - 1) 2, 2 d - 1 d (d - 1), 2 d - 5 d 2 - 3 d + 1和2 d - 3 d (d - 2)的平面曲线的最小日志规范阈值度d⩾3,我们描述了平面曲线度d的日志规范阈值这些数字。作为一个应用，我们证明了2d, 2d - 3 (d - 1) 2, 2d - 1 d (d - 1)， 2d - 5 d 2 - 3 d + 1和2d - 3 d (d - 2)是度为d大于或等于3的p3光滑表面的Tian的α -不变量的最小值。我们还证明，对于PGL 3 (C)组的作用而言，其对数规范阈值小于52d的每个度d小于或等于4的简化平面曲线是git不稳定的，并且我们用对数规范阈值52d描述了git半稳定的简化平面曲线。
{"title":"Worst Singularities of Plane Curves of Given Degree.","authors":"Ivan Cheltsov","doi":"10.1007/s12220-017-9762-y","DOIUrl":"https://doi.org/10.1007/s12220-017-9762-y","url":null,"abstract":"<p><p>We prove that <math> <mrow><mfrac><mn>2</mn> <mi>d</mi></mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <msup><mrow><mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mn>2</mn></msup> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>1</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>5</mn></mrow> <mrow><msup><mi>d</mi> <mn>2</mn></msup> <mo>-</mo> <mn>3</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn></mrow> </mfrac> </mrow> </math> and <math> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo></mrow> </mfrac> </math> are the smallest log canonical thresholds of reduced plane curves of degree <math><mrow><mi>d</mi> <mo>⩾</mo> <mn>3</mn></mrow> </math> , and we describe reduced plane curves of degree <i>d</i> whose log canonical thresholds are these numbers. As an application, we prove that <math> <mrow><mfrac><mn>2</mn> <mi>d</mi></mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <msup><mrow><mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> <mn>2</mn></msup> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>1</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo></mrow> </mfrac> <mo>,</mo> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>5</mn></mrow> <mrow><msup><mi>d</mi> <mn>2</mn></msup> <mo>-</mo> <mn>3</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn></mrow> </mfrac> </mrow> </math> and <math> <mfrac><mrow><mn>2</mn> <mi>d</mi> <mo>-</mo> <mn>3</mn></mrow> <mrow><mi>d</mi> <mo>(</mo> <mi>d</mi> <mo>-</mo> <mn>2</mn> <mo>)</mo></mrow> </mfrac> </math> are the smallest values of the <math><mi>α</mi></math> -invariant of Tian of smooth surfaces in <math> <msup><mrow><mi>P</mi></mrow> <mn>3</mn></msup> </math> of degree <math><mrow><mi>d</mi> <mo>⩾</mo> <mn>3</mn></mrow> </math> . We also prove that every reduced plane curve of degree <math><mrow><mi>d</mi> <mo>⩾</mo> <mn>4</mn></mrow> </math> whose log canonical threshold is smaller than <math><mfrac><mn>5</mn> <mrow><mn>2</mn> <mi>d</mi></mrow> </mfrac> </math> is GIT-unstable for the action of the group <math> <mrow><msub><mi>PGL</mi> <mn>3</mn></msub> <mrow><mo>(</mo> <mi>C</mi> <mo>)</mo></mrow> </mrow> </math> , and we describe GIT-semistable reduced plane curves with log canonical thresholds <math><mfrac><mn>5</mn> <mrow><mn>2</mn> <mi>d</mi></mrow> </mfrac> </math> .</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"27 3","pages":"2302-2338"},"PeriodicalIF":1.1,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9762-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37028600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-01Epub Date: 2017-02-28DOI: 10.1007/s12220-017-9763-x
Lev Birbrair, Maciej P Denkowski
This paper is devoted to the study of the medial axes of sets definable in polynomially bounded o-minimal structures, i.e. the sets of points with more than one closest point with respect to the Euclidean distance. Our point of view is that of singularity theory. While trying to make the paper self-contained, we gather here also a large bunch of basic results. Our main interest, however, goes to the characterization of those singular points of a definable, closed set , which are reached by the medial axis.
{"title":"Medial Axis and Singularities.","authors":"Lev Birbrair, Maciej P Denkowski","doi":"10.1007/s12220-017-9763-x","DOIUrl":"https://doi.org/10.1007/s12220-017-9763-x","url":null,"abstract":"<p><p>This paper is devoted to the study of the <i>medial axes</i> of sets definable in polynomially bounded o-minimal structures, i.e. the sets of points with more than one closest point with respect to the Euclidean distance. Our point of view is that of singularity theory. While trying to make the paper self-contained, we gather here also a large bunch of basic results. Our main interest, however, goes to the characterization of those singular points of a definable, closed set <math><mrow><mi>X</mi> <mo>⊂</mo> <msup><mrow><mi>R</mi></mrow> <mi>n</mi></msup> </mrow> </math> , which are reached by the medial axis.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"27 3","pages":"2339-2380"},"PeriodicalIF":1.1,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-017-9763-x","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37029463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-01Epub Date: 2016-06-30DOI: 10.1007/s12220-016-9729-4
Andrew Homan, Hanming Zhou
Let (M, g) be an analytic, compact, Riemannian manifold with boundary, of dimension . We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in M, satisfying the Bolker condition (in: Quinto, Proceedings of conference "Seventy-five Years of Radon Transforms", Hong Kong, 1994). Using analytic microlocal analysis, we prove a microlocal regularity theorem for generalized Radon transforms on analytic manifolds defined on an analytic family of hypersurfaces. We then show injectivity and stability for an open, dense subset of smooth generalized Radon transforms satisfying the Bolker condition, including the analytic ones.
{"title":"Injectivity and Stability for a Generic Class of Generalized Radon Transforms.","authors":"Andrew Homan, Hanming Zhou","doi":"10.1007/s12220-016-9729-4","DOIUrl":"https://doi.org/10.1007/s12220-016-9729-4","url":null,"abstract":"<p><p>Let (<i>M</i>, <i>g</i>) be an analytic, compact, Riemannian manifold with boundary, of dimension <math><mrow><mi>n</mi> <mo>≥</mo> <mn>2</mn></mrow> </math> . We study a class of generalized Radon transforms, integrating over a family of hypersurfaces embedded in <i>M</i>, satisfying the Bolker condition (in: Quinto, Proceedings of conference \"Seventy-five Years of Radon Transforms\", Hong Kong, 1994). Using analytic microlocal analysis, we prove a microlocal regularity theorem for generalized Radon transforms on analytic manifolds defined on an analytic family of hypersurfaces. We then show injectivity and stability for an open, dense subset of smooth generalized Radon transforms satisfying the Bolker condition, including the analytic ones.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"27 2","pages":"1515-1529"},"PeriodicalIF":1.1,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-016-9729-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"36847503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-09-18DOI: 10.1007/s12220-014-9541-y
M. A. Hernández Cifre, J. Yepes Nicolás
{"title":"On Brunn–Minkowski-Type Inequalities for Polar Bodies","authors":"M. A. Hernández Cifre, J. Yepes Nicolás","doi":"10.1007/s12220-014-9541-y","DOIUrl":"https://doi.org/10.1007/s12220-014-9541-y","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"26 1","pages":"143 - 155"},"PeriodicalIF":1.1,"publicationDate":"2014-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-014-9541-y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52802826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2014-03-21DOI: 10.1007/s12220-014-9474-5
B. Drinovec Drnovšek
{"title":"A Disc Formula for Plurisubharmonic Subextensions in Manifolds","authors":"B. Drinovec Drnovšek","doi":"10.1007/s12220-014-9474-5","DOIUrl":"https://doi.org/10.1007/s12220-014-9474-5","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"25 1","pages":"1401 - 1408"},"PeriodicalIF":1.1,"publicationDate":"2014-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-014-9474-5","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52802298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-11-06DOI: 10.1007/s12220-013-9467-9
G. Ólafsson, J. Wolf
{"title":"The Paley–Wiener Theorem and Limits of Symmetric Spaces","authors":"G. Ólafsson, J. Wolf","doi":"10.1007/s12220-013-9467-9","DOIUrl":"https://doi.org/10.1007/s12220-013-9467-9","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"24 1","pages":"1 - 31"},"PeriodicalIF":1.1,"publicationDate":"2013-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-013-9467-9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52802250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-04-10DOI: 10.1007/s12220-013-9419-4
A. Isaev, D. Zaitsev
{"title":"Reduction of Five-Dimensional Uniformly Levi Degenerate CR Structures to Absolute Parallelisms","authors":"A. Isaev, D. Zaitsev","doi":"10.1007/s12220-013-9419-4","DOIUrl":"https://doi.org/10.1007/s12220-013-9419-4","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"23 1","pages":"1571 - 1605"},"PeriodicalIF":1.1,"publicationDate":"2013-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-013-9419-4","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52802236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2013-04-01DOI: 10.1007/S12220-011-9266-0
D. Bartolucci
{"title":"On the Best Pinching Constant of Conformal Metrics on $mathbb {S}^{2}$ with One and Two Conical Singularities","authors":"D. Bartolucci","doi":"10.1007/S12220-011-9266-0","DOIUrl":"https://doi.org/10.1007/S12220-011-9266-0","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"23 1","pages":"855-877"},"PeriodicalIF":1.1,"publicationDate":"2013-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S12220-011-9266-0","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52793546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-03DOI: 10.1007/s12220-012-9340-2
C. Demeter, Francesco Di Plinio
{"title":"Logarithmic Lp Bounds for Maximal Directional Singular Integrals in the Plane","authors":"C. Demeter, Francesco Di Plinio","doi":"10.1007/s12220-012-9340-2","DOIUrl":"https://doi.org/10.1007/s12220-012-9340-2","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"24 1","pages":"375 - 416"},"PeriodicalIF":1.1,"publicationDate":"2012-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s12220-012-9340-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52801800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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