Families of Galois closure curves for plane quintic curves (Q661377)

Let \(\Gamma\) be a smooth plane curve of degree \(d\geq 4\) over the complex numbers. Let us consider the field extension of degree \(d-1\) induced by the projection of \(\Gamma\) from one of its points \(P\) onto a line \(\ell_P\) not passing through \(P\). We define the Galois closure curve \(\Gamma_P\) as the non-singular model of the Galois closure of the above field extension. The author analyses the general properties of such a construction and focuses on the case of curve of degree \(5\). In this case it is possible to construct a smooth surface \(Y\subset \mathbb P^2\times (\mathbb P^2)^\vee\) and a \(\mathcal S_4\) cover \(X\longrightarrow Y\) whose minimal resolution \(S\) is of general type and satisfies the inequality \(c_1^2(S)>2c_2(S)\).