Minimal models of covering spaces of \(\mathbb{P}^ 2\) branched along two nonsingular curves with normal crossings (Q1176705)

Let \(\overline X\) be a minimal model of the desingularization of the finite covering \(\Psi:X\to P^ 2\), branched along two nonsingular curves \(S_ 1\) and \(S_ 2\) with only normal crossings. In this paper, the author studies the dependence of \(\overline X\) on \((n_ 1,n_ 2)=(\deg(S_ 1),\deg(S_ 2))\). --- The main result is the following: (1) if \(n_ 1+n_ 2\leq 3\), then \(\overline X\) is a rational surface; (2) if \(n_ 1+n_ 2=4\), then \(\overline X\) is either a rational surface, a K3 surface, or an elliptic surface; (3) if \(n_ 1+n_ 2=5\), then \(\overline X\) is either a K3 surface, or a surface of general type; (4) if \(n_ 1+n_ 2=6\), then \(\overline X\) is either K3, elliptic, or of general type; (5) if \(n_ 1+n_ 2\geq 7\), then \(\overline X\) is of general type. Moreover, if \(\overline X\) is of general type, then the index \(\tau(\overline X)\) of \(\overline X\) is a negative number.