Subsets of $\mathbb{N}$ arising as genera of smooth projective curves in a variety Given a smooth projective variety, the genera of the smooth projective curves in it form a subset of $\mathbb{N}$.
Assuming the dimension is at least $2$, I think this subset is polynomially spaced, i.e., there is a non-constant polynomial $P$ such that there is an element between $P(n)$ and $P(n+1)$ for all $n\in\mathbb{N}$.

Question. What polynomially spaced subsets of $\mathbb{N}$ do we get this way? Can we get cofinite proper subsets? Does the answer depend on the ground field?

 A: Let me give a partial answer about surfaces.
You can surely get the whole of $\mathbb{N}$, at least over $\mathbb{C}$.
Let $Q = \mathbb{P}^1 \times \mathbb{P}^1$, and let $L$, $M$ be the classes of the two rulings. Then, for $a, \, b >0$, the divisor $aL + bM$ is very ample, of genus $(a-1)(b-1)$.
In particular, the genus of $(n+1)L+2M$ is $n$.
I am not sure that it is possible to give a complete classification of all possible subsets that one obtains, since there in principle are a lot of different situations. Just to give an example, the subset $$\{n^2+1 \mid n \in \mathbb{N} \}$$ is realized by a simple, principally polarized Abelian surface.
As another example, for all $N \in \mathbb{N}$ you can realize a proper infinite subset of $\mathbb{N}$ containing the set $\{1, \ldots, N\}$ by considering suitably chosen nodal curves in $\mathbb{P}^2$ and blowing up the nodes.
A: On a surface, the adjunction formula tells us that the genus of a smooth projective curve is a quadratic polynomial of the class of the curve. The class of the curve lies in $\mathbb Z^n$, $n$ the Picard rank of the surface, so the set of genera looks like the image of a polynomial map from $\mathbb Z^n$, except for the fact that not every class necessarily contains a smooth projective curve. (For negative classes, this is true for straightforward reasons, but there can be subtler obstructions as well.)
Francesco Polizzi gives a few examples where one knows exactly which classes contain smooth curves, and thus can give a precise description. Surely one can find more examples with a little effort, but a complete classification handling all surfaces seems elusive.
