Effective bounds for semistable sheaves on a threefold (Q2422377)

The paper gives effective upper bounds for \(ch_3(E)\) of \(\mu\)-semistable torsion-free sheaves \(E\) on arbitrary projective threefold over an algebraically closed field \(k\) with char\((k)=0\). Let \(X\) be a smooth projective variety of dimension \(n\geq 2\) and \(H\) an ample line bundle on \(X\). Let \(E\) be an \(H\)-semistable torsion-free sheaf of rank \(r\geq 2\) on \(X\). By A. Langer's result, we have an explicit upper bound for \(h^0(X,E)\) depending on \(ch_1(E)\), \(r\) and \(H\). We also have \(H^{n-1}(X,E(K_X+lH))=0\) due to H. Sun, where \(K_X\) is the canonical line bundle on \(X\) and \(l\) is any integer bigger than some number depending on \(ch_1(E)\), \(ch_2(E)\), \(r\) and \(H\). Using Riemann-Roch, we get a upper bound for \(\chi(E(K_X+lH))\leq h^0(E(K_X+lH))+h^2(E(K_X+lH))=h^0(E(K_X+lH))\). Hence we get a upper bound for \(ch_3(E)\) depending on \(ch_1(E)\), \(ch_2(E)\), \(r\) and \(H\). By a similar strategy and calculations, the author gets upper bound for \(\dim \operatorname{Ext}^1(E,E)\) and \(\dim \operatorname{Ext}^3(E,E)\), hence the upper bound for the dimension and lower bound for the expected dimension of the moduli space \(M\) of semistable sheaves at the point \([E]\) corresponding to a \(\mu\)-stable locally free sheaf \(E\).