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Suppose you are given a smooth quartic surface $X$ in $\mathbb P^3$. I would like to find an upper bound for the number of lines on $X$ in the case that there is no plane intersecting the curves in four lines (so that the possible intersections of the surface and a plane are an irreducible quartic curve, a line and an irreducible cubic, or two lines and a smooth conic).

One possible way of finding such a bound is the following.

Suppose you are given a triangle-free graph $\Gamma$. Form the matrix $M_\Gamma = (m_{ij})$ in this way: set $m_{ii} = -2$, and $m_{ij} = 1$ or $0$ ($i \neq j$) if the vertices $v_i$ and $v_j$ are connected respectively not connected by an edge. In other words, $M_\Gamma$ is $A_\Gamma - 2I$, where $A_\Gamma$ is the adjacency matrix of the graph $\Gamma$. Note that they have the same eigenvalues shifted by two.

Of course, the vertices of the graph represent the lines on the surface and they are connected if and only if the lines intersect. Taking into account the Lefschetz bound on the Picard number, we can ask: if the rank of the matrix $M_\Gamma$ is lower than 20 (in the complex case) or 22 (in the positive characteristic case), how many vertices can the graph $\Gamma$ have?

The question without the "triangle-freeness" hypothesis seems also interesting to me.

Before plunging in the theory of algebraic graph theory, I would like to ask you whether this approach has already been pursued. Any reference to articles or books related to this problem (for example, on nice characterizations of graphs admitting the eigenvalue 2) will also be appreciated.

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    $\begingroup$ I cannot comment specifically about the problem with your hypotheses. However, in general, the bounds implied by considering Betti numbers / Picard number are far worse than the bounds obtained by "enumerative methods". You might look at my appendix to an article of Browning and Heath-Brown on this subject. $\endgroup$
    – Jason Starr
    May 13, 2014 at 9:06
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    $\begingroup$ Ok, I found the article. It's "The density of rational points on non-singular hypersurfaces", II, Proc. London Math. Soc. 93 (2006), 273-303. $\endgroup$
    – Davide Cesare Veniani
    May 13, 2014 at 11:43

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Yes, this approach has been tried, and we're about to submit a paper [edit: the paper has now been submitted, see arXiv:1601.04238]. Alas, very little is known about hyperbolic (as we call them; those with a single positive eigenvalue) graphs, and currently the proof is heavily computer aided (too many cases) and still using some algebraic geometry (each line gives rise to an elliptic pencil, and these pencils are studied arithmetically). Accidentally, just the Picard number estimate is not enough: one also has to use more subtle criteria of embeddability of a lattice to $2E_8\oplus3U$.

Segre's bound is $64$ in general (there is a gap in the proof) and $48$ in your case (no plane fully split; no gap). For the moment, see arXiv:1303.1304 for further details and modern proof.

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  • $\begingroup$ Your paper does not seem to mention hyperbolic graphs. Is there some other place where the connection is written up? $\endgroup$
    – Felix Goldberg
    May 14, 2014 at 9:36
  • $\begingroup$ I said we are about to submit a paper. The paper I referred to uses the classical approach and bridges the gap in Segre's proof. $\endgroup$
    – Alex Degtyarev
    May 14, 2014 at 14:18
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    $\begingroup$ I guess it is worth mentioning this article, again by Rams and Schütt, where the proof on the bound of 64 lines can be found arXiv:1212.3511. This is actually the article that inspired my question. $\endgroup$
    – Davide Cesare Veniani
    May 14, 2014 at 20:57
  • $\begingroup$ @AlexDegtyarev I couldn't track down your bound of 48 lines in neither of the articles mentioned. Segre treats this case at the end of his article, giving a bound of 64. Rams and Schütt in arXiv:1212.3511 give a similar proof in Lemma 5.5, producing a bound of 62. $\endgroup$
    – Davide Cesare Veniani
    May 14, 2014 at 21:07
  • $\begingroup$ Oops. Sorry, actually, that's the one I meant. As to Segre, let me double check. $\endgroup$
    – Alex Degtyarev
    May 15, 2014 at 4:05
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A while ago I had a long discussion of something along these lines (although the setting was somewhat different, IIRC) with colleagues, who are, unlike me, experts on K3 things. First of all, there is a bound on the number of lines on quartics due to B.Segre. Another outcome of the discussion was that there is a connection with combinatorics of generalised quadrangles (GQs) with lines of size 4. It turned out that some examples give rise to the (famous in combinatorics) examples of GQ(3,5) and GQ(3,9), with, respectively 64 and 112 points (giving the respective configurations of lines -- it escapes me now whether the points or the lines of GQs correspond to the lines on surfaces). Details escape me now. I'll email this question to these colleagues, perhaps they can comment more.

Regarding the graph eigenvalues question, there is a lot of literature. If the minimal eigenvalue of the adjacency matrix is at least -2, then you have a full classification of such graphs. See e.g. the book by Brouwer and Haemers.

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