This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Pinch point" mathematics – news · newspapers · books · scholar · JSTOR(May 2022)
Section of the Whitney umbrella, an example of pinch point singularity.
The equation for the surface near a pinch point may be put in the form
where [4] denotes terms of degree 4 or more and is not a square in the ring of functions.
For example the surface near the point , meaning in coordinates vanishing at that point, has the form above. In fact, if and then {} is a system of coordinates vanishing at then is written in the canonical form.
The simplest example of a pinch point is the hypersurface defined by the equation called Whitney umbrella.
The pinch point (in this case the origin) is a limit of normal crossings singular points (the -axis in this case). These singular points are intimately related in the sense that in order to resolve the pinch point singularity one must blow-up the whole -axis and not only the pinch point.
pinch, point, mathematics, this, article, relies, largely, entirely, single, source, relevant, discussion, found, talk, page, please, help, improve, this, article, introducing, citations, additional, sources, find, sources, pinch, point, mathematics, news, new. This article relies largely or entirely on a single source Relevant discussion may be found on the talk page Please help improve this article by introducing citations to additional sources Find sources Pinch point mathematics news newspapers books scholar JSTOR May 2022 In geometry a pinch point or cuspidal point is a type of singular point on an algebraic surface Section of the Whitney umbrella an example of pinch point singularity The equation for the surface near a pinch point may be put in the form f u v w u 2 v w 2 4 displaystyle f u v w u 2 vw 2 4 where 4 denotes terms of degree 4 or more and v displaystyle v is not a square in the ring of functions For example the surface 1 2 x x 2 y z 2 0 displaystyle 1 2x x 2 yz 2 0 near the point 1 0 0 displaystyle 1 0 0 meaning in coordinates vanishing at that point has the form above In fact if u 1 x v y displaystyle u 1 x v y and w z displaystyle w z then u v w displaystyle u v w is a system of coordinates vanishing at 1 0 0 displaystyle 1 0 0 then 1 2 x x 2 y z 2 1 x 2 y z 2 u 2 v w 2 displaystyle 1 2x x 2 yz 2 1 x 2 yz 2 u 2 vw 2 is written in the canonical form The simplest example of a pinch point is the hypersurface defined by the equation u 2 v w 2 0 displaystyle u 2 vw 2 0 called Whitney umbrella The pinch point in this case the origin is a limit of normal crossings singular points the v displaystyle v axis in this case These singular points are intimately related in the sense that in order to resolve the pinch point singularity one must blow up the whole v displaystyle v axis and not only the pinch point See also editWhitney umbrella Singular point of an algebraic varietyReferences editP Griffiths J Harris 1994 Principles of Algebraic Geometry Wiley Classics Library Wiley Interscience pp 23 25 ISBN 0 471 05059 8 Retrieved from https en wikipedia org w index php title Pinch point mathematics amp oldid 1216580341, wikipedia, wiki, book, books, library,
