Let $X$ be a smooth cubic surface in $\mathbb{P}^3$. It is a classical theorem of Cayley and Salmon that $X$ contains exactly 27 lines over an algebraically closed field. 


In 2002, Heath-Brown proved in his paper "The density of rational points on curves and surfaces" that if $F$ is a binary form of degree $d \geq 3$ which is not a perfect $d$th power, then the only lines on $X: F(x\_1, x\_2) = F(x\_3, x\_4)$ correspond to $F(x\_1, x\_3) = F(x\_3, x\_4) = 0$, i.e., the line corresponds to a pair of roots of $F$, or there exists $a\_1, a\_2, a\_3, a\_4$ such that $F(x\_1, x\_2) = F(a\_1 x\_1 + a\_2 x\_2, a\_3 x\_1 + a\_4 x\_2)$, i.e., the line corresponds to an automorphism of the binary form $F$.


Now let $F$ be a smooth cubic form (i.e., its discriminant is non-zero). Then the surface $X : F(x\_1, x\_2) - F(x\_3, x\_4) = 0$ is a smooth cubic surface, hence it should have 27 lines on it. However, I count 9 lines coming from roots ($3^2$ many) and $6$ coming from automorphisms of $F$ (the automorphism group of $F$ in $\operatorname{GL}\_2(\mathbb{C})$ is always isomorphic to $D\_3$, the dihedral group on 3 letters). This only gives 15 lines, so there are 12 missing lines. By Heath-Brown's theorem, the only way to account for the discrepancy seems to be to count certain lines with multiplicity. I am not sure which are the ones that need to be counted with multiplicity. 


Any help would be much appreciated.



A question regarding lines on a cubic surface
To each automorphism of the binary form there correspond actually $d$ lines, not only one. In case $d = 3$, since there are always 6 automorphisms, we get $6\cdot 3 = 18$ lines, which summed to the 9 lines coming from roots give a total of 27.


This is very well explained in [this article](http://arxiv.org/abs/math/0606100) (Theorem 3.1):



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> S. Boissière and A. Sarti, *Counting lines on surfaces*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. VI(2007), 39-52.
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Curious fact: the 27 lines on a smooth cubic are indeed always distinct. If one allows the surface to have isolated rational double points, then the number is always strictly smaller, but there is a "natural" way to assign a multiplicity to each line such that the sum of the multiplicities is still equal to 27, see the [following article](http://jlms.oxfordjournals.org/content/s2-19/2/245.extract):



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> J. W. Bruce and C. T. C. Wall, *On the classification of cubic surfaces*, J. London Math. Soc. (1979) s2-19 (2): 245-256.
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