Abstract
We find the complete integer solutions of the equation
X
2
+
Y
2
+
Z
2
-
4
X
Y
-
4
Y
Z
+
10
X
Z
=
1
X^{2}+Y^{2}+Z^{2}-4XY-4YZ+10XZ=1
.
As an application, we prove that, for each solution
(
a
,
b
,
c
)
(a,b,c)
such that
a
>
0
a>0
,
b
-
2
a
>
0
b-2a>0
and
(
b
-
2
a
)
2
≥
4
a
(b-2a)^{2}\geq 4a
, there is a vector bundle 𝐸 on
P
3
\mathbb{P}^{3}
defined by a minimal linear resolution
0
→
O
P
3
(
-
2
)
a
→
O
P
3
(
-
1
)
b
→
O
P
3
c
→
E
→
0
0\to\mathcal{O}_{\mathbb{P}^{3}}(-2)^{a}\to\mathcal{O}_{\mathbb{P}^{3}}(-1)^{b}\to\mathcal{O}_{\mathbb{P}^{3}}^{c}\to E\to 0
.
In particular, 𝐸 satisfies
χ
(
End
E
)
=
1
\chi(\operatorname{End}E)=1
.
ON THE CHOW THEORY OF PROJECTIVIZATIONS
