Singularities of Chow varieties

Question

Let $X$ be a smooth complex projective variety. The Chow variety of degree $d$, $r$ dimensional subvarieties is denoted by $C_{d,r}(X)$. The Chow variety can have many topologically connected components (The number of connected components can grow to infinity as $d$ goes to infinity). Each connected component can be a reducible variety with irreducible components of different dimensions. There is a natural monoidal structure on the Chow variety. We call a complex point on $C_{d,r}(X)$ indecomposable if it does not lie in the image of $C_{d_1,r}(X)\times C_{d_2,r}(X)$ for any $d_1>0$ and $d_2>0$ where $d_1+d_2=d$ under this monoidal operation. The indecomposable points form a Zariski open subset of complex points of each irreducible component. For example when $r=0$, for $d>1$ there are no indecomposable points. Indecomposable points exist only when $d=1$ and it is the $X$ itself. I have two questions:

1. Are the Zariski open subvariety of indecomposable points of irreducible components of $C_{d,r}(X)$ necessarily smooth? If not how bad the singularity can be? Are they local complete intersections?

2. Define the indecomposable dimension of $C_{d,r}(X)$ as the minimum of dimensions of irreducible components that happen to consist of only indecomposable points, if there are no such components then we set it to infinity (for example when $r=0$ there is only $X$ in degree $1$ and for $d>1$ this dimension is infinite). Does dimension grow to infinity as $d$ goes to infinity?

Answer 1

I am just posting my comments as an answer.

Question 1 is false. Here is a variant of my comment. Consider the dense open subscheme of the Chow variety parameterizing degree-$3$ curves in projective $n$-space that are (geometrically) irreducible and reduced. This has one irreducible component that is a dense open subscheme of the total space of a projective space bundle of relative dimension $9$ over the Grassmannian parameterizing $2$-planes in the projective $n$-space. The fibers parameterize the space of irreducible plane cubics in the corresponding $2$-plane. So the total dimension of this irreducible component equals $3(n-2) + 9 = 3n+3$.

There is a second irreducible component that has a dense open subscheme that is a dense open in the total space of a Zariski locally trivial fiber bundle of relative dimension $12$ over the Grassmannian of $3$-planes. The fibers parameterize linearly nondegenerate, irreducible and reduced, degree-$3$ curves of arithmetic genus-$0$ in the corresponding $3$-plane. Since the Grassmannian parameterizing $3$-planes in projective $n$-space has dimension $4(n-3)$, the total dimension of this irreducible component equals $4(n-3)+12 = 4n$.

These two irreducible components intersect along the locus parameterizing nodal plane cubics. This locus is nonempty, irreducible, and has dimension $3n+2$. Every local complete intersection scheme, indeed every Cohen-Macaulay scheme, is pure-dimensional. Thus, when $n\geq 4$, the Chow variety parameterizing degree-$2$ curves in projective $n$-space is not locally a complete intersection, it is not Gorenstein, it is not Cohen-Macaulay, and it is not even pure-dimensional since it has two intersection components of differing dimension $3n+3$ and $4n$.

Question 2 is false. The blowing up of the projective plane along the base locus of a very general pencil of plane cubics is a surface that admits a morphism to the projective line (the "base" of the "pencil") whose geometric fibers are at-worst-nodal, integral curves of arithmetic genus equal to $1$. However, there are infinitely many $(-1)$-curves in this surface. The $9$ base points give $9$ cross-sections of the morphism to the projective line. Since this is a genus-$1$ fibration, we have an action on the smooth locis of the fibers by the degree-$0$ relative Picard. The differences between pairs of the cross-sections gives a free Abelian subgroup of the relative Picard of rank $8$, and this countable group acts on the set of $(-1)$-curves to give a countably infinite collection of $(-1)$-curves. Each of these curves gives an isolated point of the Chow variety of curves (or the Hilbert scheme, or the space of stable maps, etc.). By quasi-compactness of the components of the Chow variety with fixed degree (or the Hilbert scheme with fixed Hilbert polynomial, etc.), only finitely many of these isolated points can lie in a single Chow variety of fixed degree. So there are infinitely many positive degrees such that the corresponding Chow variety has an isolated point.