# Two infinite series of moduli spaces of rank 2 sheaves on $${\mathbb {P}}^3$$ P 3

Research paper by Marcos Jardim, Dimitri Markushevich; Alexander S. Tikhomirov

Indexed on: 17 Jan '17Published on: 05 Jan '17Published in: Annali di Matematica Pura ed Applicata (1923 -)

#### Abstract

Abstract We describe new components of the Gieseker–Maruyama moduli scheme $${\mathcal {M}}(n)$$ of semistable rank 2 sheaves E on $${\mathbb {P}^{3}}$$ with $$c_1(E)=0$$ , $$c_2(E)=n$$ and $$c_3(E)=0$$ whose generic point corresponds to nonlocally free sheaves. We show that such components grow in number as n grows, and discuss how they intersect the instanton component. As an application, we prove that $${\mathcal {M}}(2)$$ is connected, and identify a connected subscheme of $${\mathcal {M}}(3)$$ consisting of seven irreducible components.AbstractWe describe new components of the Gieseker–Maruyama moduli scheme $${\mathcal {M}}(n)$$ of semistable rank 2 sheaves E on $${\mathbb {P}^{3}}$$ with $$c_1(E)=0$$ , $$c_2(E)=n$$ and $$c_3(E)=0$$ whose generic point corresponds to nonlocally free sheaves. We show that such components grow in number as n grows, and discuss how they intersect the instanton component. As an application, we prove that $${\mathcal {M}}(2)$$ is connected, and identify a connected subscheme of $${\mathcal {M}}(3)$$ consisting of seven irreducible components. $${\mathcal {M}}(n)$$ $${\mathcal {M}}(n)$$E $${\mathbb {P}^{3}}$$ $${\mathbb {P}^{3}}$$ $$c_1(E)=0$$ $$c_1(E)=0$$ $$c_2(E)=n$$ $$c_2(E)=n$$ $$c_3(E)=0$$ $$c_3(E)=0$$n $${\mathcal {M}}(2)$$ $${\mathcal {M}}(2)$$ $${\mathcal {M}}(3)$$ $${\mathcal {M}}(3)$$