Given $n, k, d$, let $\mathrm{Chow}(n, k, d)$ be the Chow variety parameterizing algebraic cycles of pure dimension $k$ and degree $d$ in $\mathbb{P}^n$. It is a projective subvariety of $\mathbb{P}(H^0(\mathcal{O}_G(d)))$, where $G$ denotes the Grassmannian $G(n-k-1, n)$ embedded in $\mathbb{P}^{{n+1\choose n-k}-1}$ under the Plücker embedding. My main question is: what upper bounds do we know for the degree of $\mathrm{Chow}(n, k, d)$ in $\mathbb{P}(H^0(\mathcal{O}_G(d)))$?
The only upper bound I have seen is stated in the following paper:
Fabrizio Catanese: Chow varieties, Hilbert schemes, and moduli spaces of surfaces of general type.
The upper bound stated there is $3^\lambda$, where $\lambda={{n+d\choose d}\choose k+1}-1$ is an upper bound for $\dim \mathbb{P}(H^0(\mathcal{O}_G(d)))$. See Theorem (1.30) of the paper. By the way, I think we should be able to exchange $k$ with $n-k-1$ by exchanging primal and dual Plücker coordinates. Also, I think we can also choose $\lambda={{n+1\choose n-k}+d\choose d}-1$ as this is also an upper bound for $\dim \mathbb{P}(H^0(\mathcal{O}_G(d)))$, which follows simply from the fact $G\subseteq \mathbb{P}^{{n+1\choose n-k}-1}$.
When $k\in [\epsilon n, (1-\epsilon)n]$ for some constant $\epsilon>0$, the bound $3^\lambda$ is doubly-exponential in $n$. This seems to be far from being tight, as noted in Remarks (1.31) of the above paper: when $d=1$, the Chow variety is simply a Grassmannian under the Plücker embedding and we know its degree is only singly-exponential.
I wonder if there are better bounds known now and what is the correct bound we should expect. In particular, I would be happy if there is a singly-exponential bound when $d$ is bounded.
