# Zero schemes of sections of locally free sheaves

Let $$X$$ be a scheme. Let $$\mathcal E$$ be a locally free sheaf of rank $$r$$ on $$X$$ and let $$s$$ be a section of $$\mathcal E$$. Then the zero scheme of $$s$$ is defined as follows: Consider the homomorphism $$\mathcal O_X \to \mathcal E$$ induced by $$s$$, taking duals, we obtain $$\mathcal E^\vee \to \mathcal O_X$$. Then $$Z(s)$$ is defined to be the scheme associated to the sheaf of ideals $$\mathop{\mathrm{im}}(\mathcal E^\vee \to \mathcal O_X)$$.

First question: is there some natural conditions of regularity defined on $$s$$. For example, when $$\mathcal E = \mathcal L$$ is an invertible $$\mathcal O_X$$-module, then $$s$$ is said to be regular if and only if the induced homomorphism $$\mathcal O_X \to \mathcal L$$ is injective, and in this case $$Z(s)$$ is an effective Cartier divisor on $$X$$. So I think the condition of regularity should satisfy that if $$s$$ is regular, then any generic point of irreducible components of $$Z(s)$$ has codimension $$r$$ in $$X$$.

Now suppose we have already defined some conditions of regularity. Assume now that $$X$$ is a smooth projective variety and let $$s$$ be a "regular" section. Consider the $$r$$-cycle associated to $$Z(s)$$. Prove the following statement: The class of the $$r$$-cycle associated to $$Z(s)$$ in the $$r$$-th Chow group $$\mathop{\mathrm{CH}}^r(X)$$ equals the $$r$$-th Chern class $$c_r(\mathcal E)$$ of $$\mathcal E$$.

Hence the linear equivalence class of $$Z(s)$$ is independent of the choice of $$s$$ and we get a well-defined map $$\{\text{locally free sheaves of rank } r \text{ on } X\} / \{\text{isomorphisms}\} \to \mathop{\mathrm{CH}}\nolimits^r(X),$$ is this map bijective? (When $$r = 1$$, we obtain an isomorphism $$\mathop{\mathrm{Pic}}(X) \to \mathop{\mathrm{Cl}}(X)$$.)

If the map above is bijective, then the group structure on $$\mathop{\mathrm{CH}}\nolimits^r(X)$$ should induces a group structure on $$\{\text{locally free sheaves of rank } r \text{ on } X\} / \{\text{isomorphisms}\}$$, and what is it? (When $$r = 1$$, it is tensor products of invertible $$\mathcal O_X$$-modules. However, we cannot simply take the tensor product of two locally free sheaves of rank $$r$$).

• Locally $s$ is given a collection of functions $f_1,f_2\ldots$. $s$ would be regular if the sequence is regular in the usual sense of commutative algebra. Concerning your 3rd paragraph, it's a bit more complicated. See p 151 of Grothendieck, La theorie des classes de Chern. The last paragraph seems pretty far off. The closest correct statement involves the relation between $K_0(X)$ and $CH^*(X)$. Nov 22, 2020 at 17:11
• Of course the local description globalises to a definition in terms of the Koszul complex: a section $\mathscr E^\vee \to \mathcal O$ is regular if and only if the Koszul complex $0 \to \wedge^r \mathscr E^\vee \to \ldots \to \mathscr E^\vee \to \mathcal O$ is exact. Nov 22, 2020 at 21:13

The question of what regularity means was addressed in the comments, and I don't have any more to add. In particular, if $$s$$ is a regular section of a rank $$r$$ bundle $$\mathcal{E}$$, the class of $$Z(s)$$ in the Grothendieck group $$K_0(X)$$ is $$\sum (-1)^i [\wedge^i \mathcal{E}^\vee]$$. When $$X$$ is a nonsingular variety, $$r$$th Chern class of this is $$(-1)^r(r-1)![Z(s)]$$, where $$[Z(s)]$$ is the class in $$CH^r(X)$$. See, for example, page 151 of Grothendieck, La théorie des classes de Chern.
Finally, let me point out that the (normalized) $$r$$th Chern class generally won't give a bijection between the pointed set of isomorphism classes of rank $$r$$ bundles and $$CH^r(X)$$. To see this, let $$X=\mathbb{P}^2$$ and $$r=2$$. Then I'll leave you to check that $$c_2(T_X)= c_2(O(1)\oplus O(3))\in CH^2(X)\cong \mathbb{Z}$$ but the bundles are not isomorphic. Probably the best you can hope for along the lines you are suggesting is the isomorphism $$K_0(X)\otimes \mathbb{Q}\cong CH^*(X)\otimes\mathbb{Q}$$, when $$X$$ is a nonsingular variety.