Geography for surfaces of general type in positive characteristic (Q1191367)

The problem of geography is to deduce various properties of a minimal surface \(X\) of general type (for example, its existence or uniruledness) from constraints on its Chern numbers \((c_ 2, c_ 1^ 2)\). The aspect of this problem considered here is the following: Fix a positive real number \(\alpha\). Then in the \((x,y)\)-plane with \(x = c_ 2\) and \(y = c^ 2_ 1\), consider three regions: \(A = A_ \alpha\) given by \(c^ 2_ 1 \leq \alpha c_ 2\); \(B = B_ \alpha\) given by \(0 \leq \alpha c_ 2 < c^ 2_ 1\); \(C\) given by \(c_ 2 < 0\). It is well known that in characteristic \(p>0\), each of these regions is populated (i.e. there exist surfaces whose Chern numbers lie in these regions) while in characteristic 0, \(B_ 3\) and \(C\) are empty. Our main results are these: Theorem 7. Any surface with \(c_ 2<0\) is uniruled. (This was conjectured by Raynaud.) Theorem 10. A surface with \(c_ 2 = 0\) is purely inseparably covered by one of special type. Corollary 15. If \(c^ 2_ 1 (X) > [16p^ 2/(4p^ 2-1)]\cdot c_ 2 (X)\), then \(\Omega^ 1_ X\) is unstable (in the sense of Bogomolov). Theorem 16. If \(\alpha = \alpha (p) = {5 \over 4} (p + {1 \over p} + 2)\), then any \(B\)-surface \(X\) (i.e. a surface in region \(B = B_ \alpha)\) is purely inseparably covered by a surface \(Y\) that is either an \(A\)-surface with \(c^ 2_ 1 (Y) < c^ 2_ 1 (X)\) or is of special type. We work throughout over an algebraically closed field \(k\) of characteristic \(p>0\).