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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 \({\mathcal {M}}(2)\) \({\mathcal {M}}(2)\) \({\mathcal {M}}(3)\) \({\mathcal {M}}(3)\)