5
$\begingroup$

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.

$\endgroup$
  • 6
    $\begingroup$ The answer cannot depend on multiplicities. The 27 lines on a smooth cubic surface are always distinct. $\endgroup$ – Francesco Polizzi Apr 6 '16 at 13:19
  • 3
    $\begingroup$ 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. $\endgroup$ – Francesco Polizzi Apr 6 '16 at 13:22
  • 1
    $\begingroup$ I couldn't find the result you (mis)quote in Heath-Brown's paper. Could you give a precise reference? $\endgroup$ – abx Apr 6 '16 at 13:53
  • 1
    $\begingroup$ @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. $\endgroup$ – Stanley Yao Xiao Apr 6 '16 at 14:00
  • 2
    $\begingroup$ Maybe Theorem 3.1 in Boissière-Sarti might help. $\endgroup$ – Davide Cesare Veniani Apr 6 '16 at 15:08
2
$\begingroup$

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 (Theorem 3.1):

S. Boissière and A. Sarti, Counting lines on surfaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. VI(2007), 39-52.

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:

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.