On rank 3 quadratic equations of projective varieties
Authors: Park Euisung 
Published: November 2023

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma left-parenthesis upper X comma upper L right-parenthesis equals StartSet left-parenthesis upper A comma upper B right-parenthesis bar upper A comma upper B element-of upper P i c left-parenthesis upper X right-parenthesis comma upper L equals upper A squared circled-times upper B comma h Superscript 0 Baseline left-parenthesis upper X comma upper A right-parenthesis greater-than-or-equal-to 2 comma h Superscript 0 Baseline left-parenthesis upper X comma upper B right-parenthesis greater-than-or-equal-to 1 EndSet period">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi 3 left-parenthesis upper X comma upper L right-parenthesis equals union Underscript left-parenthesis upper A comma upper B right-parenthesis element-of normal upper Sigma left-parenthesis upper X comma upper L right-parenthesis Endscripts upper W left-parenthesis upper A comma upper B right-parenthesis">

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.