= Solution
The dense <algebraic torus> has coordinate ring $k[M]$, a <Laurent polynomial ring> and hence a <unique factorization domain>. Its <divisor class group> is therefore zero. Restrict an arbitrary <Weil divisor> $D$ to the <algebraic torus> and choose a <rational function on an algebraic variety> $f$ whose divisor there is $D|_T$. Then $D-\operatorname{div}(f)$ is supported on the <algebraic torus> boundary. The prime divisors in that boundary are precisely the $F_j$ corresponding to the <toric rays>, by the <torus orbit-cone correspondence>. Consequently
$$
\boxed{D\sim\sum_{j=1}^d a_jF_j\quad\text{for some }a_j\in\mathbb Z.}
$$
Since the variety is smooth, every <Weil divisor> is Cartier. This gives invariant representatives for all divisor classes, without assuming that the original divisor was invariant.
Now use such a representative. The associated vector space is
$$
L(D)=\{f\in k(X)^*: \operatorname{div}(f)+D\ge0\}\cup\{0\}=H^0(X,\mathcal O_X(D)).
$$
The <principal divisor on a toric variety> formula gives $\operatorname{div}(\chi^m)=\sum_j\langle m,e_j\rangle F_j$. Define the <lattice polytope of a toric divisor>
$$
P_D=\{m\in M_{\mathbb R}:\langle m,e_j\rangle\ge-a_j\text{ for every }j\}.
$$
For a maximal <toric cone> $\sigma$, its basic <toric ray> vectors are a basis of $N$, so there is a unique integral $m_\sigma$ with $\langle m_\sigma,e_j\rangle=-a_j$ on its <toric rays>. The <algebraic torus character> $\chi^{m_\sigma}$ is a local frame for $\mathcal O(D)$ on $U_\sigma$. Every global section restricts on the <algebraic torus> to a finite <Laurent polynomial> $\sum_m c_m\chi^m$. Relative to this frame, regularity on $U_\sigma$ means $\sum_m c_m\chi^{m-m_\sigma}\in k[\sigma^\vee\cap M]$. Since distinct <algebraic torus characters> are independent, this holds exactly when each exponent with nonzero coefficient satisfies the inequalities for the <toric rays> of $\sigma$. Taking all charts proves
$$
\boxed{L(D)=\bigoplus_{m\in P_D\cap M}k\chi^m.}
$$
Completeness makes $P_D$ bounded when nonempty: a nonzero recession vector would be nonnegative on every <toric ray> of a complete <toric fan>, and therefore on all of $N_{\mathbb R}$, which is impossible. The displayed space is consequently finite-dimensional.
At the torus-fixed point $p_\sigma$ in the smooth chart $U_\sigma\cong\mathbb A^n$, a section $\chi^m$ has local <monomial> $\chi^{m-m_\sigma}$. Its value is nonzero exactly when this <monomial> is constant, namely when $m=m_\sigma$. Thus if the <complete linear system of a divisor> has no base point, some section is nonzero at $p_\sigma$, forcing $m_\sigma\in P_D$. Conversely, if $m_\sigma\in P_D$ for every maximal <toric cone>, its <algebraic torus character> is a global section whose local expression on $U_\sigma$ is one. It therefore vanishes nowhere on that whole chart, and these charts cover $X$. This proves the <toric basepoint-free criterion>:
$$
\boxed{|D|\text{ is basepoint-free}\ \Longleftrightarrow\ \forall\sigma\in\Sigma(n),\ \langle m_\sigma,e_j\rangle\ge-a_j\text{ for all }j,\text{ with equality on }\sigma.}
$$
The analogous <toric ampleness criterion>, stated without proof, replaces the inequalities off $\sigma$ by strict ones:
$$
\boxed{D\text{ is ample}\ \Longleftrightarrow\ \forall\sigma\in\Sigma(n),\ \langle m_\sigma,e_j\rangle=-a_j\text{ on }\sigma,\quad\langle m_\sigma,e_j\rangle>-a_j\text{ off }\sigma.}
$$
Equivalently, the <normal fan of a polytope> $P_D$ is exactly $\Sigma$.
For the final <fan subdivision>, take $n\ge2$ so that $\tilde e$ is a new <toric ray>, as the construction requires. The vectors $e_1,\ldots,e_n$ are a lattice basis, so their sum is primitive. Replacing any one basis vector by this sum gives another basis. Thus the <star subdivision> gives a smooth <toric blowup at a torus-fixed point> $\pi:X_{\Sigma'}\to X_\Sigma$, with <algebraic exceptional divisor> $E=\tilde F$. On the old chart, a local equation of $D$ is $\chi^{-m_\sigma}$, whose order on the new <toric ray> is $-\langle m_\sigma,\tilde e\rangle=\sum_{i=1}^n a_i=a$. All old <toric ray> coefficients are unchanged. Therefore
$$
\boxed{D'=\pi^*D.}
$$
We prove ampleness of $D_c=cD'-E$ directly by inequalities, giving the <ample divisor after blowing up a torus-fixed point> result.
Let $u_1,\ldots,u_n$ be the basis of $M$ dual to $e_1,\ldots,e_n$. On the new maximal <toric cone> $\sigma_i$, which omits $e_i$, the required local <algebraic torus character> is
$$
m_{i,c}=cm_\sigma+u_i.
$$
Indeed its pairings on $e_j$, $j\ne i$, are $-ca_j$, while its pairing on $\tilde e$ is $-ca+1$, exactly the negative of the new coefficient $ca-1$. Its inequality on the omitted old <toric ray> has strict gap one:
$$
\langle m_{i,c},e_i\rangle+ca_i=1.
$$
For any <toric ray> $e_j$ outside the original <toric cone>, put $\delta_j=\langle m_\sigma,e_j\rangle+a_j$. The original ampleness gives positive integers $\delta_j$, and the new strict gaps are
$$
\langle m_{i,c},e_j\rangle+ca_j=c\delta_j+\langle u_i,e_j\rangle>0
$$
for all sufficiently large $c$.
For any unchanged maximal <toric cone> $\eta\ne\sigma$, the local <algebraic torus character> is $cm_\eta$. All strict inequalities on old <toric rays> follow from those for $D$. Its inequality on the new <toric ray> has gap
$$
\langle cm_\eta,\tilde e\rangle+ca-1=c\epsilon_\eta-1,\qquad\epsilon_\eta=\sum_{i=1}^n(\langle m_\eta,e_i\rangle+a_i).
$$
Each summand is nonnegative, and at least one is positive because $\eta\ne\sigma$ cannot contain all its <toric rays>. Hence $\epsilon_\eta$ is a positive integer, making this gap positive for $c\ge2$. There are only finitely many remaining inequalities. For example, any integer satisfying
$$
c>\max\left\{1,\ \max_{\substack{1\le i\le n\\j>n}}\frac{-\langle u_i,e_j\rangle}{\delta_j}\right\}
$$
works, with the inner maximum omitted when its index set is empty. The <toric ampleness criterion> now gives
$$
\boxed{cD'-\tilde F\text{ is ample for every sufficiently large integer }c.}
$$
The new-ray assumption is essential to the printed construction. In dimension one, $\tilde e=e_1$ and the <fan subdivision> is unchanged, so the two purported distinct divisors coincide. If the displayed definition of $D'$ is applied literally on $\mathbb P^1$ with coefficients $a_1=-2$, $a_2=3$, then $D$ has degree one but $D'$ has degree $-1$, contradicting the conclusion. The argument above applies to the intended blowup in dimension at least two.
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