# Cusp

From Encyclopedia of Mathematics

*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 |

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Cusp.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cusp&oldid=34164