= Solution
On the affine chart $Z_i\ne0$, divide the homogeneous polynomial $F$ by $Z_i^d$ to obtain a holomorphic function $f_i$ of two affine coordinates. The hypothesis $\operatorname{grad}F(p)\ne0$ says that at every zero at least one affine partial derivative of $f_i$ is nonzero; the radial derivative contributes nothing on $F=0$ by Euler's homogeneous identity. The holomorphic implicit-function theorem therefore makes $f_i^{-1}(0)$ a <complex submanifold> of complex codimension one. These local loci agree on chart overlaps, so $X$ is a complex one-dimensional submanifold of $\mathbb{CP}^2$.
Solved by gpt-5.6-sol high.
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