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on Asymptotic line-count constant for smooth surfaces of degree $d$ in $\mathbb{P}^3$ in characteristic $0$ ($C_{76}$)

The statement

For each integer $d \ge 3$ , let $\ell_0(d)$ denote the maximal number of lines contained in a smooth surface of degree $d$ in $\mathbb P^3_{\mathbb C}$ .

We define

C_{76} := \limsup_{d\to\infty}\frac{\ell_0(d)}{d^2}.

The constant $C_{76}$ measures the quadratic growth rate of the maximal line count on smooth complex degree- $d$ surfaces. The problem is open; the best established range is

3 \le C_{76} \le 11.

[BS2007-lb-3d2] [BR2023-ub-11d2]

The current best general upper bound is due to Bauer and Rams,

\ell_0(d) \le 11d^2 - 30d + 18,

while a standard infinite lower-bound family comes from surfaces of the form $\varphi(x,y)-\psi(z,t)=0$ , which give at least $3d^2$ lines for every $d$ . [BR2023-ub-11d2] [BS2007-lb-3d2]

Bauer and Rams also note that the exact fixed-degree maximum remains unknown for every $d \ge 5$ . [BR2023-ub-11d2]

What gets accepted

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