Deformations of covers, Brill-Noether theory, and wild ramification (Q2566561)

In the paper under review, the author studies deformations of a map between two smooth curves, with partially prescribed branching. The main theorem is as follows. Theorem. Given \(C\) and \(D\) smooth curves over a field \(k\), and \(f: C \rightarrow D\) of degree \(d\), together with \(k\)-valued points \(P_1, \ldots, P_n\) of \(C\) such that \(f\) is ramified to order at least \(e_i\) at each \(P_i\) for some \(e_i\), then the space of first-order infinitesimal deformations of \(f\) together with the \(P_i\) over \(k\), such that \(f(P_i)\) remains fixed and the \(P_i\) remain ramified to order at least \(e_i\) is parameterized by \[ H^0\biggl(C, f^* T_D\Bigl(- \sum_i(e_i - \delta_i) P_i \Bigr)\biggr)\oplus k^{\sum_i(1 - \delta_i)} \] where \(\delta_i = 0\) if \(p | e_i\) or \(f\) is ramified to order higher than \(e_i\) at \(P_i\) and is 1 otherwise. Furthermore, the space of first-order infinitesimal deformations of \(C\), the \(P_i\) and \(f\), fixing \(f(P_i)\) and preserving the ramification condition at each \(P_i\), is parameterized by \[ \mathbb{H}^1\biggl(C, T_C \Bigl(-\sum_i P_i\Bigr) \rightarrow f^*T_D\Bigl(-\sum_i e_i P_i\Bigr)\biggr) \cong k^{d(2 - 2 g_D) - (2 - 2 g _C) - \sum_i (e_i - 1)} \] where the last isomorphism requires also that \(f\) be separable. Finally, both statements still hold when some of the \(e_i\)'s are allowed to be 0, which we interpret to put no condition on the \(P_i\) at all. After proving the theorem, the author uses it to get a positive-characteristic Brill-Noether type result for ramified maps from general curves to the projective line, which holds also for wild ramification indices. The paper ends with some more results on wild ramification, in the case where \(C = D = \mathbb{P}^1\).