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Mathematics, Algebraic Geometry
Abstract:
We describe new irreducible components of the moduli space of rank $2$
semistable torsion free sheaves on the three-dimensional projective space whose
generic point corresponds to non-locally free sheaves whose singular locus is
either 0-dimensional or consists of a line plus disjoint points. In particular,
we prove that the moduli spaces of semistable sheaves with Chern classes
$(c_1,c_2,c_3)=(-1,2n,0)$ and $(c_1,c_2,c_3)=(0,n,0)$ always contain at least
one rational irreducible component. As an application, we prove that the number
of such components grows as the second Chern class grows, and compute the exact
number of irreducible components of the moduli spaces of rank 2 semistable
torsion free sheaves with Chern classes $(c_1,c_2,c_3)=(-1,2,m)$ for all
possible values for $m$; all components turn out to be rational. Furthermore,
we also prove that these moduli spaces are connected, showing that some of
sheaves here considered are smoothable.
