The moduli space M(r,n) of framed torsion free sheaves on the projective plane with rank r and second Chern class equal to n has the natural action of the (r+2)-dimensional torus. In this paper, we look at the fixed point set of different one-dimensional subtori in this torus. We prove that in the homogeneous case the generating series of the numbers of the irreducible components has a beautiful decomposition into an infinite product. In the case of odd r, these infinite products coincide with certain Virasoro characters. We also propose a conjecture in a general quasihomogeneous case.
We describe new irreducible components of the Gieseker-Maruyama moduli scheme M(3) of semistable rank 2 coherent sheaves with Chern classes c1=0, c2=3, c3=0 on P^3, general points of which correspond to sheaves whose singular loci contain components of dimensions both 0 and 1. These sheaves are produced by elementary transformations of stable reflexive rank 2 sheaves with c1=0, c2=2 along a disjoint union of a projective line and a collection of points in P^3. The constructed families of sheaves provide first examples of irreducible components of the Gieseker-Maruyama moduli scheme such that their general sheaves have singularities of mixed dimension.
A new irreducible component of the Gieseker–Maruyama moduli scheme M(3) of semistable coherent sheaves of rank 2 with Chern classes c1=0,c2=3, and c3 = 0 on P3 such that its general point corresponds to a sheaf whose singular locus contains components of dimensions 0 and 1 is described. These sheaves are obtained by elementary transformations of stable reflexive sheaves of rank 2 with Chern classesc1=0,c2=2, and c3 = 2 along the projective line. The constructed family of sheaves is the first example of an irreducible component of a Gieseker–Maruyama scheme whose general point corresponds to a sheaf with singularities of mixed dimension.