Condition for singular points in $F_{\mu} =X^3+Y^3+Z^3+ \mu XYZ$ in $\Bbb{P}^{2}_{\Bbb{C}}$?

Solution 1:

Your curve should be defined by $ F_{\mu} = x^3 + y^3 + z^3 + 3 \mu xyz $. In that case, first observe that for $ \mu = 0 $ you get a nonsingular curve. So assume $ \mu \neq 0 $ and let $ [a:b:c] $ be a singular point with $ a \neq 0 $ WLOG. The equation $ 3a^2 = -3 \mu bc $ shows then that $ b,c $ are nonzero. We also have $ b^2 = - \mu ca $ and $ c^2 = - \mu ab $ and multiplying these three equalities and cancelling out $ a^2b^2c^2 $ gives $ 1 = - \mu^3 $.

Conversely, suppose $ \mu^3 = -1 $. Then we check straightaway that $ [-1: \mu : \mu ] $ is a singular point.