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''ordinary cusp''
 
''ordinary cusp''
  
A singular point of specific type of an algebraic curve. Namely, a singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027430/c0274301.png" /> of an algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027430/c0274302.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027430/c0274303.png" /> is called a cusp if the completion of its local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027430/c0274304.png" /> is isomorphic to the completion of the local ring of the plane algebraic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027430/c0274305.png" /> at the origin.
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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"> W. Fulton,   "Algebraic curves. An introduction to algebraic geometry" , Benjamin (1969) pp. 66</TD></TR></table>
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<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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[[Category:Algebraic geometry]]

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