I wish to ask the naive question: if we write down a random curve $C$ over $\mathbb{F}_q$, what can be said about the probability that $C(\mathbb{F}_q)=\emptyset$?
Of course, this depends on the family from which we pick $C$. The application I have in mind seems very ad hoc (written below), but it will be very nice to learn any insight towards the question in general! I suppose the same question can be asked for any reasonable family of varieties too.
For my application, for any odd $n$ and odd $q=p^r$ consider all monic separable polynomials $f(x)\in\mathbb{F}_q[x]$ of degree $n$. Let $\bar{C}$ be the smooth completion of $(y^2=f(x))$, a hyperelliptic curve. Consider $i:\bar{C}\hookrightarrow\operatorname{Jac}(\bar{C})$ by $p\mapsto (p)-(\infty)$. Pick a random $D'\in\operatorname{Jac}(\bar{C})(\mathbb{F}_q)$ and consider $f_{D'}:\operatorname{Jac}(\bar{C})\twoheadrightarrow\operatorname{Jac}(\bar{C})$ given by $D\mapsto 2D+D'$. Consider $C:=\bar{C}\times_{\operatorname{Jac}(\bar{C})}\operatorname{Jac}(\bar{C})$ given by $i$ and $f_{D'}$. I am interested in the probability for $C(\mathbb{F}_q)=\emptyset$.
Let me mention that the above construction have significant similarity with well-established works regarding arithmetic statistics of hyperelliptic curves over global fields; in fact I first learned of the above construction from such works. My side of the story is that $C(\mathbb{F}_q)=\emptyset$ happens to be the criterion to get exceptional structure in the ``wave-front set'' of certain representations of $p$-adic groups, which in a nutshell is related to some vast generalization of Fourier coefficients for modular forms. I was thinking about these regarding my work in https://arxiv.org/abs/2207.13445.
