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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

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    $\begingroup$ Intuitively, if you fix the degree and assume the curve is nondengerate in a projective space of some dimension, and then increase the dimension, I would assume that the genus of your curve decreases (this is related to the Castelnuovo bound). E.g., A cubic curve in $\Bbb{P}^2$? Elliptic curve. A cubic curve in $\Bbb{P}^3$? Twisted cubic. Of course, weird things happen over finite fields but if the field is large enough I think the intuition would hold. $\endgroup$
    – Eoin
    Aug 17 '20 at 20:52
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One way to get some intuition comes from looking at the (weaker) combinatorial bound. Suppose you had a nondegenerate curve $C$ in some projective space $\mathbb P^n$. Suppose that that $L$ is a subspace of codimension $2$ in $\mathbb P$ and that $|C\cap L|=m$. The higher the dimension $n$ gets, the higher value we are allowed to pick for $m$. Indeed we can always find at least $n-1$ points in $C$ that span a $\mathbb P^{n-2}$.

Bezout tells you that for any hyperplane $H$ that contains $L$, the number of points of $C$ that lie in $H$ and don't lie in $L$ is at most $d-m$. Since the number of such hyperplanes is $q+1$, independent of the dimension, we get $|C|-m\le (q+1)(d-m)$ or equivalently from rearranging terms $$|C|\le (d-m)q+d.$$ For $m=n-1$ this gives the bound $|C|\le (d-n+1)q+d$ for all nondegenerate curves $C$. Of course this is weaker than the conjecture and theorems that you mention in the post, but (1) it holds true for all curves including the one that violates the Sziklai bound (2) it already exhibits the phenomenon "bound gets tighter as $n$ goes up".

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  • $\begingroup$ Thanks for your answer Gjergji. I'd like to leave the question up just a bit longer to see if it attracts any other discussion, but this does help. $\endgroup$
    – TomGrubb
    Aug 18 '20 at 16:43
  • $\begingroup$ Is this really much weaker? It seems only a little bit weaker to me. $\endgroup$
    – Will Sawin
    Aug 18 '20 at 17:50
  • $\begingroup$ @WillSawin I changed it to just say weaker. The papers above do some work to really refine this bound and settle the conjecture in dimensions $2,3$. So the problem gets harder the higher $n$ (and therefore $d$) gets, but that's where this bound starts getting worse. $\endgroup$ Aug 18 '20 at 17:58

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