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What is the definition of nodal singularity of an algebraic curve ?

I got the following definition from here:

A nodal singularity of an algebraic curve is one of the forms parameterized by the equation $xy=0$. A nodal curve is a curve with a nodal singularity.

Apparently, it is not clear to me the parametrization $xy=0$.

Can you please explain it?

MAS
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1 Answers1

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It means that the completion of the local ring at the point is isomorphic to $k[[x, y]]/(xy)$. Intuitively, if you zoom way in it looks like the letter $X$ at the bad point.

hunter
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  • So, If we consider an algebraic curve in $n$-variables $x_1,~x_2,~,\cdots,x_n$, then the completion of the local ring at the nodal point is isomorphic to the ring $K[[x_1,~x_2, \cdots, x_n]]/(x_1x_2 \cdots x_n)$. Is it ? – MAS Jul 13 '20 at 14:46
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    @mathvision since it's a curve, it should be one-dimensional, so the isomorphism class is always $k[[x, y]]/(xy)$ (the number of variables in the completion is just two, regardless of the dimension of the ambient space in which the curve is cut out). – hunter Jul 13 '20 at 15:26