[*general hyperplane section*]{} of $S$.

For a fixed scheme $S$, a general hyperplane section is proper. Namely, the set $\mathcal V$ of hyperplanes in ${\mathbb{P}}^{n+1}$ that do not contain any component of $S$ is open and nonempty.

If the scheme $S$ has dimension $d\geq 1$, then a general hyperplane section has dimension $d-1$. If $I_S$ is the homogeneous saturated ideal of the scheme $S\subseteq{\mathbb{P}}^{n+1}$, then $I_{S\cap H|H}=H^0_*({\mathbb{P}}^n,{\mathcal{I}}_{S\cap H})\subseteq R/(H)$ is the homogeneous saturated ideal of the general hyperplane section $S\cap H\subseteq H$. The following short exact sequence of ideal sheaves relates $S$ to a general hyperplane section $S\cap H$ $$0{\longrightarrow}{\mathcal{I}}_S(-1)\stackrel{\cdot H}{{\longrightarrow}}{\mathcal{I}}_S{\longrightarrow}{\mathcal{I}}_{S\cap H|H}{\longrightarrow}0.$$ Taking cohomology we get the exact sequence: $$\xymatrix{0\ar[r] & I_S(-1)\ar[r]^{\cdot H} & I_S\ar[rr]^{\pi}\ar[dr]& & I_{S\cap H|H}
\ar[r] & H^1_*({\mathcal{I}}_S)(-1)\\
 & & & I_S+(H)/(H)\ar[ur]\ar[dr] & & \\
 & & 0\ar[ur] & & 0 & }$$ If $S$ is arithmetically Cohen-Macaulay, or more in general if $R/I_S$ has depth at least 2, then $H^1_*({\mathcal{I}}_S)=0$, and $I_{S\cap H|H}=I_S+(H)/(H).$ In particular, if $S$ is arithmetically Cohen-Macaulay then a proper hyperplane section of $S$ is also arithmetically Cohen-Macaulay. Recall that a scheme $S$ of dimension $d\geq 0$ is arithmetically Cohen-Macaulay if and only if $H^i_*({\mathcal{I}}_S)=0$ for $1\leq i\leq d$. Every zero-dimensional scheme is 