Uniruled surfaces of general type
The author studies surfaces \(S\) in positive characteristics arising from the following construction: Let \(C,F\) be two curves endowed with \(p\)-closed rational vector fields \(\delta_C,\delta_F\), take the quotient of \(C\times F\) by the resulting rational vector field \(\delta_C+\delta_F\), and let \(S\) be a resolution of singularities. This construction is used to show that surfaces with unusual cohomological properties exist in abundance. The author mainly concentrates on uniruled surfaces of general type in characteristic two. For example, it is shown that there are families of such surfaces with the same reduced Picard scheme, but with \(h^{1,0}\) or \(h^{0,1}\) arbitrarily large, and that there are unbounded families of such surfaces having a purely inseparable canonical map factoring over a rational surface, or having nontrivial global vector fields. Furthermore, the author constructs a surface of general type with \(\chi=1\) and \(c_1^2=14\), which violates the Bogomolov-Miyaoka-Yau inequality \(c_1^2\leq 9\chi\).
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