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Let $\mathcal E$ be a locally free sheaf of rank $e$ on a smooth projective variety $X$. Let $s$ be a global section. I want to understand:

What is the closed zero subscheme $Z(s)$ of $s$?

As a set, this is the locus of points $x \in X$ such that the image of $s$ in $\mathcal E_{x}/\mathcal m_x \mathcal E_{x}$ is zero (as described here). Then in the answer, Justin Campbell gives a description of what the scheme structure on the locus should be. In particular, we should have multiplicities associated to the irreducible components of $Z(s)$. Unfortunately, I have only a weak understanding of schemes. Surely this is more naturally understood in terms of schemes, but is there a perhaps ad hoc way of defining this in the language of varieties?

If it helps at all, I am trying to understand degeneracy loci of $i$ general sections of $\mathcal E$. Say $s_1, \dots, s_i$ are $i$ general global sections of $\mathcal E$, and consider the global section $u=s_1 \wedge \dots \wedge s_i$ of $\bigwedge^i \mathcal E$. The irreducible components of $D(u)$ have codimension $\leq e+1-i$, and for general choices of $s_1, \dots, s_i$, the codimension is exactly $e+1-i$. So $D(u)$ will yield a cycle in $A^{e+1-i}(X)$. I want to understand how the multiplicities are assigned to the $(e+1-i)$-cycles, in the language of varieties. A reference for this can be found here.

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    $\begingroup$ Giving a section $s$ of a vector bundle $E$ is same as a morphism of sheaves $s:\mathcal{O}_X\to E$ and thus induces the dual map $E^*\to\mathcal{O}_X$. So, the image of this map is an ideal sheaf and the scheme defined by this ideal sheaf is called $Z(s)$. In general, for the definition at least, it is better to work with schemes. $\endgroup$ – Mohan Oct 23 '16 at 18:10
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In the most naive terms you have the following. Locally on some open set your vector bundle is trivial and vanishing of section $s$ is vanishing of all its components $s_i(x)$ (think about sections as locally vector functions). So locally you have $n$ equations, cutting a subvariety in $X$.

Because these local "vector functions" are not independent, but a local presentation of a global sections all these local set of zeroes could be glued together into a global subvariety.

Now understanding closed zero subscheme $Z(s)$ of $s$ is understanding what is the structure sheaf of this subvariety. Because it is possible to have several components, different order of vanishing of $s$ on different components and this information must be the subscheme structure.

Returning to local description on some open set $U$, the map dual to the section $E^* \to \mathcal{O}_X$ locally sends some $(x_1, x_2, \dots x_n) \in \Gamma(U,E^*)$ to $\sum_i x_i s_i$. The image is an ideal in the ring $\mathcal{O}_X(U)$ generated by components $s_1,s_2 \dots s_n$ of the section $s$. So $\mathcal{O}_X(U)$ modulo this ideal are functions on the subvariety and this is the structure sheaf that we put on this subvariety when we talk about "closed zero subscheme".

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