Difference between revisions of "Cusp"
From Encyclopedia of Mathematics
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''ordinary cusp'' | ''ordinary cusp'' | ||
| − | A singular point of specific type of an algebraic curve. Namely, a singular point | + | A singular point of specific type of an algebraic curve. Namely, a singular point $x$ of an algebraic curve $X$ over an algebraically closed field $k$ is called a cusp if the completion of its local ring $\mathcal O_{X,x}$ is isomorphic to the completion of the local ring of the plane algebraic curve $y^2+x^3=0$ at the origin. |
====Comments==== | ====Comments==== | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Fulton, "Algebraic curves. An introduction to algebraic geometry" , Benjamin (1969) pp. 66 {{MR|0313252}} {{ZBL|0681.14011}} </TD></TR></table> |
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Latest revision as of 19:23, 1 November 2014
ordinary cusp
A singular point of specific type of an algebraic curve. Namely, a singular point $x$ of an algebraic curve $X$ over an algebraically closed field $k$ is called a cusp if the completion of its local ring $\mathcal O_{X,x}$ is isomorphic to the completion of the local ring of the plane algebraic curve $y^2+x^3=0$ at the origin.
Comments
A cusp can also be defined via the so-called intersection number of two plane curves at a point, cf. [a1], pp. 74-82. A generalization of a cusp is a hypercusp, cf. [a1], p. 82.
References
| [a1] | W. Fulton, "Algebraic curves. An introduction to algebraic geometry" , Benjamin (1969) pp. 66 MR0313252 Zbl 0681.14011 |
How to Cite This Entry:
Cusp. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cusp&oldid=18769
Cusp. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cusp&oldid=18769