Let \(S\) be a projective surface, defined over a field of arbitrary characteristic, with polarization \(H\). The moduli space of rank \(r\), slope \(H\)-semistable, torsion free sheaves on \(X\) needs not to be smooth or even reduced. Nevertheless, it is known that in characteristic \(0\), for any fixed determinant \(L\in\) Pic(\(S\)), and for \(c_2\gg 0\), the moduli space \(M_{L,c_2}\) of stable sheaves with Chern classes \((L,c_2)\) is generically smooth and irreducible. The author extends the previous result to arbitrary characteristic, providing at the same time a computable constant \(C=C(L,H)\) such that \(M_{L,c_2}\) is reduced irreducible for any \(c_2\geq C\).
The main tool for proving the result, after a careful setting of the deformation theory of sheaves in positive characteristic, relies on a sharp estimate of the Castelnuovo-Mumford regularity of a sheaf \(E\), i.e. of the first twist \(E(nH)\) whose first cohomology group vanishes. This in turn comes from a bound on the dimension of the space of global sections of a sheaf, via the restriction to a general curve \(D\in | H| \). A bound of this sort is indeed obtained for the case \(S=\mathbb P^2\), then extended to arbitrary \(S\), using the pullback of a projection \(S\to\mathbb P^2\). Numerically, the result on the constant \(C\) comes from a sharp estimate of the dimension of subvarieties in \(M_{L,c_2}\) parameterizing sheaves whose level of stability (the Segre invariant) is bounded.