# A question regarding lines on a cubic surface

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.

• The answer cannot depend on multiplicities. The 27 lines on a smooth cubic surface are always distinct. Apr 6, 2016 at 13:19
• Anyway, just check your argument on the Fermat cubic, where $$F(u,v)=u^3+v^3,$$ and see explicitly where your hidden lines lie. Apr 6, 2016 at 13:22
• I couldn't find the result you (mis)quote in Heath-Brown's paper. Could you give a precise reference?
– abx
Apr 6, 2016 at 13:53
• @abx the exact journal reference is: D.R. Heath-Brown, "The density of rational points on curves and surfaces", Annals of Mathematics (2) 155, 2002, 553-598. Apr 6, 2016 at 14:00
• Maybe Theorem 3.1 in Boissière-Sarti might help. Apr 6, 2016 at 15:08

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.