Let
X
⊂
P
r
X \subset \mathbb {P}^r
be a linearly normal variety defined by a very ample line bundle
L
L
on a projective variety
X
X
. Recently it is shown by Kangjin Han, Wanseok Lee, Hyunsuk Moon, and Euisung Park [Compos. Math. 157 (2021), pp. 2001–2025] that there are many cases where
(
X
,
L
)
(X,L)
satisfies property
Q
R
(
3
)
\mathsf {QR} (3)
in the sense that the homogeneous ideal
I
(
X
,
L
)
I(X,L)
of
X
X
is generated by quadratic polynomials of rank
3
3
. The locus
Φ
3
(
X
,
L
)
\Phi _3 (X,L)
of rank
3
3
quadratic equations of
X
X
in
P
(
I
(
X
,
L
)
2
)
\mathbb {P}\left ( I(X,L)_2 \right )
is a projective algebraic set, and property
Q
R
(
3
)
\mathsf {QR} (3)
of
(
X
,
L
)
(X,L)
is equivalent to that
Φ
3
(
X
)
\Phi _3 (X)
is nondegenerate in
P
(
I
(
X
)
2
)
\mathbb {P}\left ( I(X)_2 \right )
.
In this paper we study geometric structures of
Φ
3
(
X
,
L
)
\Phi _3 (X,L)
such as its minimal irreducible decomposition. Let
Σ
(
X
,
L
)
=
{
(
A
,
B
)
∣
A
,
B
∈
P
i
c
(
X
)
,
L
=
A
2
⊗
B
,
h
0
(
X
,
A
)
≥
2
,
h
0
(
X
,
B
)
≥
1
}
.
\begin{equation*} \Sigma (X,L) \!=\! \{ (A,B) \mid A,B \!\in \! {Pic}(X),~L \!=\! A^2 \otimes B,~h^0 (X,A) \!\geq \! 2,~h^0 (X,B) \!\geq \! 1 \}. \end{equation*}
We first construct a projective subvariety
W
(
A
,
B
)
⊂
Φ
3
(
X
,
L
)
W(A,B) \subset \Phi _3 (X,L)
for each
(
A
,
B
)
(A,B)
in
Σ
(
X
,
L
)
\Sigma (X,L)
. Then we prove that the equality
Φ
3
(
X
,
L
)
=
⋃
(
A
,
B
)
∈
Σ
(
X
,
L
)
W
(
A
,
B
)
\begin{equation*} \Phi _3 (X,L) ~=~ \bigcup _{(A,B) \in \Sigma (X,L)} W(A,B) \end{equation*}
holds when
X
X
is locally factorial. Thus this is an irreducible decomposition of
Φ
3
(
X
,
L
)
\Phi _3 (X,L)
when
P
i
c
(
X
)
{Pic} (X)
is finitely generated and hence
Σ
(
X
,
L
)
\Sigma (X,L)
is a finite set. Also we find a condition that the above irreducible decomposition is minimal. For example, it is a minimal irreducible decomposition of
Φ
3
(
X
,
L
)
\Phi _3 (X,L)
if
P
i
c
(
X
)
{Pic}(X)
is generated by a very ample line bundle.