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Let $A$ be an element of the copositive cone $\copos{n}$. A zero $\vu$ of $A$ is a nonnegative vector whose elements sum up to one and such that $\vu^TA\vu = 0$. The support of $\vu$ is the index set $\Supp{\vu} \subset \{1,\dots,n\}$ corresponding to the nonzero entries of $\vu$. A zero $\vu$ of $A$ is called minimal if there does not exist another zero $\vv$ of $A$ such that its support $\Supp{\vv}$ is a strict subset of $\Supp{\vu}$. Our main result is a characterization of the cone of feasible directions at $A$, i.e., the convex cone $\VarK{A}$ of real symmetric $n \times n$ matrices $B$ such that there exists $\delta > 0$ satisfying $A + \delta B \in \copos{n}$. This cone is described by a set of linear inequalities on the elements of $B$ constructed from the set of zeros of $A$ and their supports. This characterization furnishes descriptions of the minimal face of $A$ in $\copos{n}$, and of the minimal exposed face of $A$ in $\copos{n}$, by sets of linear equalities and inequalities constructed from the set of minimal zeros of $A$ and their supports. In particular, we can check whether $A$ lies on an extreme ray of $\copos{n}$ by examining the solution set of a system of linear equations. In addition, we deduce a simple necessary and sufficient condition on the irreducibility of $A$ with respect to a copositive matrix $C$. Here $A$ is called irreducible with respect to $C$ if for all $\delta > 0$ we have $A - \delta C \not\in \copos{n}$.