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

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