On the 1-pointed curves arising as étale covers of the affine line in positive characteristic (Q2475960)

Let \(X\) be a smooth proper curve of genus \(g\) defined over an algebraically closed field of characteristic \(p>0\) and \(P\in X\). The paper addresses the question: when is \(X\setminus\{P\}\) a finite étale cover of the affine line. This is obtained through very elementary methods. Moreover, it is also discussed the minimal degree of the cover. Nevertheless, an interesting consequence of the results of the paper is: If the answer is positive, then \(p\) does not divide \(2g-1\) and \(X\) is not ordinary. More generally, covers of the affine line of the type \(X\setminus S\) for a finite subset \(S\) of \(X\) are naturally obtained from semi-stable reduction of covers in characteristic zero.