<s>[INST] Singular points of algebraic curve, multiplicity, ordinary? Let $C \in \mathbb{P}_2$ be the curve defined by the polynomial$$P(x, y, z) = x^2z^2 + y^2z^2 + y^4.$$Find the singular points of $C$. For each one, calculate the multiplicity and say whether it is ordinary or nonordinary.
 [/INST] We calculate$$P_x = 2xz^2,\text{ }P_y = 2yz^2 + 4y^3 =2y(z^2 + 2y^2),\text{ }P_z = 2x^2z + 2y^2z = 2z(x^2 + y^2).$$$P_x$ vanishes if $x = 0$ or $z = 0$. If $x = 0$, then $P_z$ vanishes if and only if $z = 0$ or $y = 0$. If $x = z = 0$, then $P = 0$ implies $y = 0$, which is impossible. Thus, if $x = 0$ then $y = 0$, and we find $[0, 0, 1]$ is a common zero of $P$ and its first derivatives. We have$$P(x, y, 1) = x^2 + y^2 + y^4.$$The lowest order part is $x^2 + y^2 = (x + iy)(x-iy)$, so $[0, 0, 1]$ is an ordinary double point.
If $z = 0$, then $P_y$ vanishes only if $y = 0$. We find that $[1, 0, 0]$ is a point on $C$ which is a common zero of all the first derivatives of $P$, so this is also a singular point. We have$$P(1, y, z) = z^2 + y^2z^2 + z^4.$$The lowest order part is $z^2$, which has a repeated factor, so $[1, 0, 0]$ is a nonordinary double point.
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