In the sequel, let $S$ be a scheme, and $X$ a locally of finite type algebraic space over $S$.

In his thesis ([R1-R4]), David Rydh introduces, among several others, the notion of *relative cycles* on $X\to S$, and, for integers $r\ge 0$, he defines the functor:

$$\text{Chow}_r(X/S) : (\text{Sch}/S)^{\rm opp}\to\text{Set}$$

by:

$$\text{Chow}_r(X/S)(T) := \text{Cycl}^{\rm prop}_r(X\times_ST/T)$$

where the right side is the set of equidimensional relative cycles of dimension $r$ on $X\times_ST\to T$, with proper support (see [R4]).

$\text{Chow}_r(X/S)$ is an fppf sheaf, ultimately because the fppf sheaf of divided powers of $X\to S$ is representable (see [R1]).

In [R4], representability of $\text{Chow}_r(X/S)$ is shown only in those cases when $\text{Chow}_r(X/S)$ can be proved to be isomorphic as a functor to some other functor known to be representable (e.g., Angéniol's Chow space (see [An]), when $S$ is of pure characteristic zero, and $X\to S$ is smooth and separated).

It appears that the best representability result one may extrapolate from [R4] is, therefore, as follows:

**Theorem 1.** Let $S$ be a scheme of pure characteristic zero, $X\to S$ a smooth and separated algebraic space, $r\ge 0$ an integer. Then the fppf sheaf $\text{Chow}_r(X/S)$ is represented by a separated algebraic space over $S$, locally of finite type.

The algebraic space representing $\text{Chow}_r(X/S)$ will be called *Chow space*, in the sequel.

It is easy to show, using representability of the Hilbert functor when $X\to S$ is projective, that the following holds:

**Theorem 2.** Let $S$ be a scheme of pure characteristic zero, $X\to S$ a smooth and projective algebraic space, $r\ge 0$ an integer. Then:

- $\text{Chow}_r(X/S)$ is proper.
- the Hilbert-Chow morphism: $$\text{Hilb}_r(X/S)\to\text{Chow}_r(X/S)$$ constructed in [R4], is proper.

**Questions:**

- Is it true that, under the assumptions in Theorem 2, $\text{Chow}_r(X/S)$ in $(1)$ is projective?
- Under the assumptions in Theorem 2, what are the currently known properties of the Hilbert-Chow morphism $\text{Hilb}_r(X/S)\to\text{Chow}_r(X/S)$ in $(2)$? Is it surjective/surjective on geometric points/birational?
- Upon inspecting Angéniol's proof of representability of his Chow functor, it appears to me the smoothness assumption can be removed, upon appropriately exploiting the deformation theory in vol. 1 of Illusie's thesis. Has this been done anywhere in the literature?
- What can one say, to this day, about the case when $S$ is of pure characteristic $p>0$?
- Around question $(2)$. Is representability of $\text{Chow}_r(X/S)$ known at least in the case when $S = \text{Spec}(k)$, for $k$ a field of characteristic $p>0$, and $X\to S$ a smooth projective $S$-scheme?

**References:**

[An] B. Angéniol. *Familles de cycles algébriques.* Springer.

[R1] D. Rydh. *Families of zero-cycles and divided powers: I. Representability.*

[R2] D. Rydh. *Families of zero-cycles and divided powers: II. The universal family.*

[R3] D. Rydh. *Hilbert and Chow schemes of points, symmetric products and divided powers.*

[R4] D. Rydh. *Families of cycles.*

reallyasking whether the ample invertible sheaves on the Chow variety (which are studied by Mumford in GIT, by Fogarty, by Knudsen-Mumford, ...) are restrictions of an invertible sheaf defined on Angeniol's Chow scheme. $\endgroup$nottrue. I believe that Mumford's pathological examples of smooth space curves of degree $14$ and genus $24$ give counterexamples. $\endgroup$