This question concerns some counterintuitive results (to me at least) regarding the number of points on a projective curve over a finite field. Namely, if one fixes the degree of the curve, but *increases* the dimension of the ambient projective space, one can get tighter bounds on the number of $\mathbb{F}_q$ points on the curve, despite there being a larger number of $\mathbb{F}_q$ points in the ambient space. Let me make this more precise with two examples.

Let $C\subset \mathbb{P}^n_{\mathbb{F}_q}$ be a projective curve of degree $d$. Suppose $C$ is nondegenerate in the sense that it is not contained in any smaller projective space $\mathbb{P}^k_{\mathbb{F}_q}$, $k<n$.

Work of Homma (extending work of Homma and Kim) has shown $$ \#C(\mathbb{F}_q)\leq (d-1)q+1, $$ with a single exception (up to isomorphism) over $\mathbb{F}_4$. This is the so called Sziklai bound, and is tight for $n=2$.

This bound is not tight for $n>2$; recently Beelen and Montanucci show that if $C\subset \mathbb{P}^3_{\mathbb{F}_q}$ is nondegenerate then in fact $$ \#C(\mathbb{F}_q)\leq (d-2)q+1. $$ They further conjecture than if $C\subset \mathbb{P}^n_{\mathbb{F}_q}$, the general bound should be $$ \#C(\mathbb{F}_q)\leq (d-n+1)q+1. $$

This is reminiscent of a phenomenon from work of Bucur and Kedlaya. For example: a random smooth curve in $\mathbb{P}^2_{\mathbb{F}_q}$ is expected to have $$q+1$$ points over $\mathbb{F}_q$ as its degree grows to infinity. A random complete intersection of two smooth degree $d$ surfaces in $\mathbb{P}^3_{\mathbb{F}_q}$ is expected to have $$ q+1 - \frac{q^{-2}(1+q^{-1})}{1+q^{-2}-q^{-5}} < q+1 $$ points over $\mathbb{F}_q$, again as $d\to\infty$.

These results are counterintuitive to me, as the number of points in the ambient projective space grows (exponentially) as $n$ does, so in particular it seems to me that it should be *easier* for curves to have $\mathbb{F}_q$ points when they are embedded in larger projective spaces. Does anyone have any intuition as to why the opposite should be true?

References:

Beelen and Montanucci: A bound for the number of points of space curves over finite fields

Bucur and Kedlaya: The probability that a complete intersection is smooth

Homma: A bound on the number of points of a curve in projective space over a finite field