Are Chow groups generated by local complete intersections?

Question

Let $X$ be a smooth projective variety over an algebraically closed field. The Chow group $\mathbb Q\mathrm{CH}^d(X)$ is $\mathbb Q$--linearly generated by irreducible subvarieties $Z \subseteq X$ of codimension $d$, modulo rational equivalence.

I am interested in the linear subspace of $\mathbb Q\mathrm{CH}^d(X)$ which is generated by the subvarieties $Z\subseteq X$ of codimension $d$ which are locally complete intersections, so those which are locally the zero set of exactly $d$ regular functions. Let us denote this subspace by $\mathbb Q\mathrm{CH}^d_{\mathrm{lci}}(X)$. Then the question is:

Are Chow groups generated by local complete intersections? I.e. does equality $$\mathbb Q\mathrm{CH}^d_{\mathrm{lci}}(X) = \mathbb Q\mathrm{CH}^d(X)$$ hold?

If for instance $d=1$, equality holds indeed, as $X$ is smooth. I suspect this not so in general for $d\geq 2$... but where to look for a counter example?

Answer 1

This is not an answer to the original question. Instead I will argue that the SUBRING of the Chow ring generated by lci subschemes is the whole ring.

Indeed, let us first check that Chern classes of vector bundles generate the Chow ring. Indeed, the structure sheaf of any subvariety $Z$ has a locally free resolution, hence its Chern classes (in particular the class of $Z$ itself) can be expressed as a liner combination of Chern classes of vector bundles.

So, it remains to check that Chern classes of vector bundles can be expressed as linear combinations of products of lci subschemes. Let us take a vector bundle $E$ of rank $r$. Let $O(h)$ be a very ample line bundle. Then for $n \gg 0$ the bundle $E(nh)$ is globally generated. Hence its top Chern class is represented by the zero locus of a generic section of $E(nh)$ which is lci. Considering different twists one can deduce that for any $i$ the class $c_i(E)h^{r-i}$ is represented by a linear combination of lci.

Now instead of zero loci of sections, consider degeneration schemes of morphisms $O^k \to E(nh)$. Again, the class of the degeneration scheme is $c_{r-k+1}(E(nh))$ and for generic morphism it is a smooth (and hence lci) subvariety. Taking different twists we conclude that $c_i(E)h^{r-k+1-i}$ is represented by a linear combination of lci.