the following properties.

-   $A/J \cong K[P]$, $A/L\cong K[Q]$ and $$A/(L+J)= A/\big(I_{SP} + (x_\sigma: \sigma \in SP \setminus \partial P)\big) \cong K[\partial P].$$

-   $\overline \Theta =\theta_1,\dots,\theta_{d-1}$ is an l.s.o.p. of $A/(L+J)$ and $\Theta=\theta_1,\dots,\theta_{d}$ is a common l.s.o.p. of $A/I_{SP}$, $A/J$ and $A/L$.

Recall that $\sum_{k=0}^d h_k$ is equal to the number of the facets of $P$. The next lemma shows that, to prove (5)–(7), it is enough to study the number of the facets of $SP$.

\[facets\] Suppose $\partial h_i=0$ for some $1 \leq i \leq d-2$. If the number of the facets of $SP$ is even then $\sum_{k=0}^d h_k$ is even.

By the construction of $SP$, the number of the facets of $P$ is equal to that of $SP$ minus that of $\partial P$. Then the statement follows from Theorem \[1.1\](3).

\[int1\] Let $v_1,v_2,\dots,v_d$ be distinct vertices of $SP$. Then

-   If $\partial h_i=0$ for some $1 \leq i \leq d-2$ then $x_{v_1}x_{v_2} \cdots x_{v_i} \in L+(\Theta)$.

-   If $h_j=0$ for some $1 \leq j \leq d$ then $x_{v_d} x_{v_{d-1}} \cdots x_{v_{d-j+1}} \in
    J+(\Theta)$.

Recall $A/J \cong K[P]$ and $A/L \cong K[Q]$. Since $Q$ is the cone of $\partial P$, $$\begin{aligned}
\label{XX1}
\dim_K \big( A/ \big( L+(\Theta) \big) 