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I was wondering, what if I ask the same question for a plane in $T_P (X)$ that does not pass through $P$? In the spirit of the approach you have already mentioned, if one finds a 3-plane $H'$ with $P \not\in H'$ and $H'$ is not a tangent any point on the threefold, can we assert that $X \cap H' \cap T_P(X)$ will not be union of d lines?
For application to problems which I am currently working on, I need to understand the structure of an (absolutely) irreducible cubic surface defined over $\mathbb{F}_q$ containing at least one line $\ell$ defined over $\mathbb{F}_q$. I need to especially understand the singular cubics. How would they look like when we take intersection of the surface with planes passing through the line $\ell$? So I thought understanding the isomorphism classes of cubic surfaces may help me. Could you tell me a reference of the fact that "number of isomorphism classes of cubic surfaces grows with $q$?
Hirschfeld, in his book titled "Finite projective spaces of three dimensions" classified all the quadric surfaces in $\mathbb{F}_q$. I am looking for similar results for cubic surfaces. I do care about every finite fields, one could ignore characteristic 2 though. It is fine to consider the singularities even over algebraic closure of $\mathbb{F}_q$.
Excellent! I was wondering if we could find something similar like the Hermitian surfaces as I mentioned in the question? In particular, does there exist a surface which has three skew lines contained in three distinct planes that intersect in a common line? What happens if it is given that the common line is contained in the surface?
