Let $R=K\left[x_{1}, \ldots, x_{n}\right]$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $R$ generated by a finite set of monomials $\left{x^{v_{1}}, \ldots, x^{v} q\right}$. As usual we use $x^{a}$ as abbreviation for $x_{1}^{a_{1}} \cdots x_{n}^{a_{n}}$, where $a=\left(a_{1}, \ldots, a_{n}\right)$ is in $\mathbb{N}^{n}$. The three central objects of study here are the following blowup algebras: (a) the Rees algebra
$$
R[I t]:=R \oplus I t \oplus \cdots \oplus I^{i} t^{i} \oplus \cdots \subset R[t],
$$
where $t$ is a new variable, (b) the associated graded ring
$$
\mathrm{gr}{I}(R):=R / I \oplus I / I^{2} \oplus \cdots \oplus I^{i} / I^{i+1} \oplus \cdots \simeq R[I t] \otimes{R}(R / I)
$$
with multiplication
$$
\left(a+I^{i+1}\right)\left(b+I^{j+1}\right)=a b+I^{i+j+1} \quad\left(a \in I^{i}, b \in I^{j}\right),
$$
and (c) the symbolic Rees algebra
$$
R_{s}(I):=R+I^{(1)} t+I^{(2)} t^{2}+\cdots+I^{(i)} t^{i}+\cdots \subset R[t],
$$
where $I^{(i)}$ is the ith symbolic power of $I$.
MATH4411 COURSE NOTES ：
If $r \leq n$, then $b \leq n-1$. Thus from now on we may assume $r=n+1$ and takes the simpler form
$$
(\alpha, b)=\lambda_{1}\left(v_{i_{1}}, 1\right)+\cdots+\lambda_{n+1}\left(v_{i_{n+1}}, 1\right)
$$
Consider the cone $C$ generated by $\mathcal{A}^{\prime \prime}=\left{\left(v_{i_{1}}, 1\right), \ldots,\left(v_{i_{n+1}}, 1\right)\right}$. Since $-e_{1}$ is not in $C$, using we obtain a point
$$
x_{0}=\left(1-\lambda_{0}\right)(\alpha, b)+\lambda_{0}\left(-e_{1}\right) \quad\left(0 \leq \lambda_{0}<1\right)
$$