Asymptotic line-count constant for smooth degree-$d$ surfaces in $\mathbb P^3$ in characteristic $0$

Upper bound, last-listed row

| $\ell_0(d) \le 11d^2 - 30d + 18$ | [BR2023] | Current best general upper bound in characteristic $0$, valid for $d \ge 3$. [BR2023-ub-11d2] |

exact pinned row

| $\ell_0(d) \le 11d^2 - 30d + 18$ | <a href="#BR2023">[BR2023]</a> | Current best general upper bound in characteristic $0$, valid for $d \ge 3$. <a href="#BR2023-ub-11d2">[BR2023-ub-11d2]</a> |

tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $\ell_0(d) \ge 3d^2$ | [BS2007] | Achieved by the classical family $\varphi(x,y)-\psi(z,t)=0$; within that family the exact count is $3d^2$ except for the finitely many exceptional degrees $4,6,8,12,20$. Therefore $C_{76} \ge 3$. [BS2007-lb-3d2] |

exact pinned row

| $\ell_0(d) \ge 3d^2$ | <a href="#BS2007">[BS2007]</a> | Achieved by the classical family $\varphi(x,y)-\psi(z,t)=0$; within that family the exact count is $3d^2$ except for the finitely many exceptional degrees $4,6,8,12,20$. Therefore $C_{76} \ge 3$. <a href="#BS2007-lb-3d2">[BS2007-lb-3d2]</a> |

tracked since 2026-07-24 — no movement seen yet