Maximal Subsheaves of Torsion-Free Sheaves on Nodal Curves

Abstract

Let Y be a reduced irreducible projective curve of arithmetic genus g 2 with at most ordinary nodes as singularities. For a subsheaf F of rank r , degree d of a torsion-free sheaf E of rank r,where the minimum is taken over all subsheaves of E of rank r . For a fixed r , s r defines a stratification of the moduli space U (r, d) of stable torsion-free sheaves of rank r, degree d by locally closed subsets U r ,s . We study the nonemptiness and dimensions of the strata. We show that the general element in each nonempty stratum is a vector bundle and it has only finitely many (respectively unique) maximal subsheaves of rank r for s r (r − r )(g − 1) (respectively s < r (r − r )(g − 1)). We prove that the tensor product of two general stable vector bundles on an irreducible nodal curve Y is nonspecial.

Torsion-free sheaves on nodal curves and triples

“…The aim of this article is to answer whether it is possible to apply the methods in [1] to get a structure like in [6] to describe torsion free sheaves on cuspidal curves. So our technique, as in [1], differs from the other papers on the subject as [2,3,4].…”

Existence of nontrivial logarithmic co-Higgs structure on curves

“…Proposition 3.3. For the bundle E in (7) with 2γ ≤ b s − b 1 < γ, the set of its co-Higgs structures is identified with a ∆-dimensional vector space. Now the assumption in ( 5) is simply b 1 + γ ≥ b s and let e be the last integer i such that b i > γ + b 1 .…”

Brauer Group and Birational Type of Moduli Spaces of Torsionfree Sheaves on a Nodal Curve

“…Note that the sum on the right hand side is infinite if both E and F are not locally free [7]. If at least one of E and F is locally free, then Ext i (F, E) = H i (F * ⊗ E), and the terms in the right side vanish except for i = 0 , 1.…”