Let $V$ be a projectively normal closed subvariety of some projective space over an algebraically closed field $\mathbb{K}$. Let $R$ be the local ring at the vertex of the affine cone over $V$ ($R$ is obtained by localizing the homogeneous coordinate ring $\mathbb{K}[V]$ at the maximal ideal generated by all the homogeneous elements of positive degree). Samuel showed that

the class group $\operatorname{Cl}(Spec(R))\cong \operatorname{Cl}(V)/(\text{hyperplane section})$

Consequently,

$R$ is a UFD $\Leftrightarrow$ every irreducible codimension $1$ subvariety of $V$ is cut out (scheme-theoretically) by a hypersurface of the ambient projective space.

I want to understand what does it mean by "cut out (scheme-theoretically)"? The meaning of the sentence is not clear to me.

Let $V$ be a projectively normal closed subvariety of some projective space over an algebraically closed field $\mathbb{K}$. Let $R$ be the local ring at the vertex of the affine cone over $V$ ($R$ is obtained by localizing the homogeneous coordinate ring $\mathbb{K}[V]$ at the maximal ideal generated by all the homogeneous elements of positive degree). Samuel showed that

the class group $\operatorname{Cl}(Spec(R))\cong \operatorname{Cl}(V)/(\text{hyperplane section})$

Consequently,

$R$ is a UFD $\Leftrightarrow$ every irreducible codimension $1$ subvariety of $V$ is cut out (scheme-theoretically) by a hypersurface of the ambient projective space.

I want to understand what does it mean by "cut out (scheme-theoretically)"? The meaning of the sentence is not clear to me.

I appreciate your help. Thanks!