For a smooth surface of degree $d \ge 3$ in $\mathbb{P}^3$, every component of the Noether–Lefschetz locus has codimension at least $d - 3$ (Green, Voisin) and at most $p_g = \binom{d-1}{3}$. The geometry behind the lower bound is taken from the literature; this concept compares the two bounds as functions of $d$, with $m = d - 1$ as in $GeometricGenus$.

The first two statements are the inequality $d - 3 \le p_g$ for all $d \ge 3$ and its strictness for $d \ge 5$; the third is the exact case of equality: the lower bound meets $p_g$ precisely for $d = 3$ and $d = 4$. The last statement identifies the gap $p_g - (d - 3)$, the "window" between the two bounds, with the OEIS sequence A005581 evaluated at $d - 3$: as a polynomial identity without division, $(d-1)(d-2)(d-3) - 6(d-3) = (d-4)(d-3)(d+1)$, where the right-hand side is six times $A005581(d-3) = \tfrac{(d-4)(d-3)(d+1)}{6}$. It was first observed numerically for $d \le 200$ and holds for every integer $d$.