Semistable rank 2 sheaves with singularities of mixed dimension on P3

Abstract

We describe new irreducible components of the Gieseker-Maruyama moduli scheme M(3) of semistable rank 2 coherent sheaves with Chern classes c 1 = 0, c 2 = 3, c 3 = 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 c 1 = 0, c 2 = 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.

The spectrum of torsion free sheaves on P3 and applications

“…where L is a line bundle over C with Hilbert polynomial P L (k) = 3k such that L ω C , where ω C is the canonical sheaf of C. Additionally, in [11], the authors obtained 3 irreducible components whose general sheaves have mixed singularities. Those components can also be obtained by the Theorem 13, and described by X(0, 2, 2, 2, 0), X(0, 2, 4, 3, 0) and X(0, 2, 4, 2, 1), irreducible components of M(0, 3, 0) with dimension 22, 24 and 26, respectively.…”

“…With the help of [5, Table 2.8.1], and [5, Table 2.12.2], we can compute the general spectrum of the sheaves in the components X(0, 2, 2, 2, 0), X(0, 2, 4, 3, 0) and X(0, 2, 4, 2, 1), and with the help of [5, Table 3 3.9.1] we can compute the general spectrum of the sheaves in the components T(0, 3, 2, 1), T(0, 3, 4, 2), T(0, 3, 6, 3) and T(0, 3,8,4). The general spectrum of the sheaves in the Instanton and Ein components can be computed by the display of the monads ( 10) and (11). Finally the general spectrum in the component C can be computed from the cohomology of the sheaf L (that can be obtained by the Hilbert polynomial of L and Riemann-Roch) in the sequence (12).…”

The spectrum of torsion free sheaves on $\mathbb{P}^3$ and applications

“…Additionally, in [11], the authors obtained 3 irreducible components whose general sheaves have mixed singularities. Those components can also be obtained by the Theorem 13, and they are identified with X(0, 2, 2, 2, 0), X(0, 2, 4, 3, 0) and X(0, 2, 4, 2, 1), irreducible components of M(0, 3, 0) with dimension 22, 24 and 26, respectively.…”

“…With the help of [5, Table 2 and [5, Table 3.9.1] we can compute the general spectrum of the sheaves in the components T(0, 3, 2, 1), T(0, 3, 4, 2), T(0, 3, 6, 3) and T(0, 3, 8, 4). The general spectrum of the sheaves in the Instanton and Ein components can be computed by the display of the monads ( 10) and (11). Finally the general spectrum in the component C can be computed from the cohomology of the sheaf L (that can be obtained by the Hilbert polynomial of L and Riemann-Roch) in the sequence (12).…”

Irreducible components of the moduli space of rank 2 sheaves of odd determinant on the projective space

“…With this definition in mind, a systematic way of producing examples of irreducible components of M(0, c 2 , 0) whose generic point corresponds to a torsion free sheaf with 0-dimensional and 1-dimensional singularities is given in [19]. Furthermore, the third author and Ivanov [18] constructed irreducible components of M(0, 3, 0) whose generic point corresponds to a torsion free sheaf with mixed singularities. Additionally, in a recent paper [17], Ivanov proved that M(0, 3, 0) has at least 11 irreducible components.…”