We know that for an effective divisor on a smooth projective variety there is a natural way of associating to it a scheme, in particular using the Cartier divisor. Can we do the same for higher codimension effective algebraic cycles? More precisely, let $X$ be a smooth projective variety and $Z\_c:=\sum\_i a\_iZ\_i$ be an algebraic cycle on $X$, where $Z\_i$ are integral of the same codimension and $a\_i>0$. Then,


1) Is there a natural way of associating to $Z\_c$ a scheme $Z$ such that the corresponding segre class $s(Z,X)$ (notation as in Fulton's Intersection theory, chapter $4$) is the cycle $Z\_c$?


2) Furthermore, if $Z\_{c\_{\mathrm{red}}}=\sum\_i Z\_i$ is local complete intersection subscheme in $X$, then can we find such a $Z$ which is also local complete intersection subscheme in $X$?



Schemes associated to algebraic cycles and local complete intersection
1) No, let $X$ be a plane and $Z\_i$ a point, then if $Z\_c = 2 Z\_i$, then the scehme would have to be a length $2$ subcheme supported at the point. But there are a $\mathbb P^1$ of these (the schemes defined by the ideal $(x^2,xy,y^2,ax+by)$) and there is no natural choice.


2) Locally, yes, by taking the $a\_i$th power of one of the defining equations. I'm not sure if this sort of local existence can be glued globally - because of the non-naturalness mentioned earlier, this is not clear.


