(Saturated)
An affine semigroup \(S \subseteq M\) is saturated if \[ k \in \ZZ_{>0}, \quad m \in M, \quad km \in S \implies m \in S . \] \(\Spec k[S]\) is normal exactly when \(S\) is saturated in the lattice \(\ZZ S\) it generates, and \(S_\sigma = \dualof{\sigma} \intersect M\) is saturated for every cone, since its defining inequalities are homogeneous.
(A semigroup that is not saturated)
Let \(S\) be generated by \((4,0), (3,1), (1,3), (0,4)\) inside \(M = \ZZ^2\). Then \(2 \cdot (2,2) = (4,4) = (4,0) + (0,4) \in S\), but \((2,2) \notin S\), so \(S\) is not saturated and \(\Spec k[S]\) is not normal.
(Normalization of an affine toric variety)
Given a non-normal affine toric variety \(X = \Spec k[S]\), set \(M = \ZZ S\), take any finite generating set \(S'\) of \(S\), and put \[ \sigma \da \dualof{\Cone(S')} \subseteq N_\RR . \] Then \(\dualof{\sigma} \intersect M\) is the saturation of \(S\), and \[ \Spec k[\dualof{\sigma} \intersect M] \longrightarrow X \] is the normalisation. It is finite and birational, and it is an isomorphism if and only if \(S\) is saturated.
(Non-normal toric varieties)
The toric variety \(X_\Sigma\) of a fan \(\Sigma\) is normal. A variety \(X\) containing a torus \(T\) as a dense open subset, with the action of \(T\) on itself extending to \(X\), need not be normal; if \(X\) is affine, its normalization is \(X_\sigma\) for \(\sigma=\dualof{\Cone(S')}\), with \(S'\) a finite generating set of its semigroup \(S\). The cuspidal cubic \(V(x^3 - y^2)\) contains the torus \(\ts{(t^2, t^3) \st t \in \GG_m}\), so \(S = \gens{2,3} \subseteq \ZZ\), which is not saturated because \(2 \cdot 1 \in S\) while \(1 \notin S\). The saturation is \(\NN \subseteq \ZZ\), and the normalisation is \(\AA^1 \to V(x^3 - y^2)\), \(t \mapsto (t^2, t^3)\).
Dependencies
PR-TORSMAFFSmooth affine toric varieties and why toric varieties are normaluses