Motivation:
Following Fulton's Intersection Theory, the Chern class of an arbitrary algebraic $\Bbbk$-scheme $X$ can be constructed as follows. First, define the graded by codimension abelian group $A_*(X)$ of cycles of subvarieties modulo rational equivalence. Then the Chern classes $c_i(E)$ of a vector bundle $E$ will be homogeneous elements of degree $i$ of the graded ring of endomorphisms of $A_*(X)$. If for any cycle $\alpha\in A_*(X)$ we write $c_i(E)\cap \alpha$ for the value of the endomorphism $c_i(E)$ applied to $\alpha$, then the following three conditions specify the Chern classes uniquely:
- If $E$ is a vector bundle on $X$ and $f\colon X'\to X$ is a proper morphism, then $f_*(c_i(f^*(E))\cap \alpha)=c_i(E)\cap\alpha$ for every $\alpha\in A_*(X)$ with $c_i(E)$ the $i^\text{th}$ graded component of $c(E)\colon A_*(X)\to A_*(X)$
- If $E$ is a vector bundle on $X$ and $f\colon X'\to X$ is a flat morphism, then $c_i(f^*(E))\cap f^*(\alpha)=f^*(c_i(E)\cap\alpha)$.
- If $E$ is a line bundle on a variety $X$, and $D$ a Cartier divisor on $X$ with $\mathcal O(D)\cong E$, then $c_1(E)\cap[X]=[D]$, where $[D]$ is the rational equivalence class of the Weyl divisor $\sum ord_V(D)\cdot[V]$.
These properties it turns out imply three additional properties
- $c_i(E)\cap c_j(E)\cap \alpha=c_j(E)\cap c_i(E)\cap\alpha$ for all $i$ and $j$.
- $c_i(E)=0$ for $i>$rank of $E$.
- $c(E)=c(E')c(E'')$ for exact sequences $0\to E'\to E\to E''\to 0$, where $c(E)=1+c_1(E)+c_2(E)+\dots$ is the endomorphism of $A_*(X)$ known as the total Chern class .
If we restrict our attention to closed projective subschemes $X=V\subset\mathbb P^n_\Bbbk$, then it is not hard to show using free resolutions that if $E$ is a coherent sheaf on $V$, then if $c(E)$ exists (for the coherent sheaf is not necessarily a vector bundle, so the some of the defining first three properties might fail), the second three properties imply that $c(E)\in\mathbb Z[c_1(\mathcal O_V(1))]$. Hence, the structure of $\mathbb Z[c_1(\mathcal O_V(1))]$ is of interest.
Specific Questions about $\mathbb Z[c_1(\mathcal O_V(1))]$:
It is obvious that $c_1(\mathcal O_V(1))$ is nilpotent of order the dimension $n$ of the subscheme $V$, so that $\mathbb Z[c_1(\mathcal O_v(1))]$ is a quotient of $\mathbb Z[t^{n+1}]$. It is also well-known that when $V=\mathbb P^n_\Bbbk$, that there are no further relations as $c_1(\mathcal O_\mathbb P^n(1))=H$, the class of hyperplanes.
My basic question is the following: if we are given the the homogeneous ideal $I$ of $\Bbbk[x_0,\dots,x_n]$ that embeds $V$ into $\mathbb P^n_\Bbbk$ by $V=Proj\Bbbk[x_0,\dots,x_n]/I$, how do the $k$-fold products $c_1(\mathcal O_V(1))^k\cap\alpha$ relate to one another?
This is certainly too broad of a question for arbitary closed projective subschemes, and even for projective varieties in general. In fact, I don't know the answer for any proper non-singular projecte varieties, for which by Poincare duality we know $c_1(\mathcal O_V(1))\cap\alpha$ is the same as properly intersecting $\alpha$ with $[D]$, so that the question reduces to what self-intersections with the rational equivalence class of the Weyl divisor associated to $\mathcal O_V(1)$ are (can we even compute a representative the rational equivalence class $[D]$?
