Are the plane curves
$C: X^5+Y^5+Z^5+(X^2+Y^2)Z^3 $
and
$D: X^5+Y^5+(X^2+Y^2)Z^3$
smooth or not?
It seems $D$ has only a node at $(0,0,1)$.
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I assume by "plane curves" you mean in the projective plane $\Bbb{P}^2$? – Nick Sep 23 '17 at 17:45
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Check "locally": set one of the variables (say $z$) equal to 1 to look in an affine chart, and then check the "Jacobian criterion". Look at the Jacobian/gradient $dC = \left( \frac{\partial C}{\partial x}, , \frac{\partial C}{\partial y}\right)$, and make sure it is not the zero vector for any of the points on the curve. – Nick Sep 23 '17 at 17:52
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Thanks Nick for the process. I checked D is a nodal curve as you said. But just wanted to make sure of the curve D if it is smooth if I am not wrong. – Rahul Sep 23 '17 at 18:08
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If it has a node, then it is not smooth. A "node" is by definition a singularity. – Nick Sep 23 '17 at 18:18