Vector bundles 𝐸 on ℙ3 with homological dimension 2 and 𝜒(End 𝐸) = 1

Abstract

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 .