Locally trivial families of hyperelliptic curves: the geometry of the Weierstrass scheme (Q2566738)

The authors consider the following problem: characterize all pairs \((C,p)\) which have constant moduli, where \(C\) is a plane curve and \(p \in {\mathbb P}^2_K\). This means that \((C,p)\) are such that if \(m= \text{mult}_p C\) and \(\deg C = d+m\), \(d\geq 3\), then for every two lines \(L_1,L_2\) containing \(p\), with \((C-\{p\})\cap L_i= \{p_{i,1},\dots,p_{i,d}\}\), \(i=1,2\), we can number the \(p_{i,j}\)'s in such a way that all the lines \(p_{1,j}p_{2,j}\) are concurrent. The main result is that if char\(K =0\), then \((C,p)\) has constant moduli if and only if there exists a birational map \(\varphi\) on \({\mathbb P}^2_K\) (which is an automorphism if \(m=0\)) and a form \(H(Y,Z)\) such that \(\varphi(p)=(1:0:0)\), almost every line \(l\) through \(p\) is such that \(\varphi (l)\) is a line through \((1:0:0)\) and \(\overline{\varphi (C)}\) has equation: \[ \Pi_{t=1}^{d/k}(X^kZ^{n-k}-\alpha_tH(Y,Z) \quad \quad \text{or } \quad X\Pi_{t=1}^{(d/1)/k}(X^kZ^{n-k}-\alpha_tH(Y,Z), \] where \(k\in {\mathbb N}\) divides either \(d\) or \(d-1\) and \(\alpha_t\in \overline{K}\), \(\forall t\). This results allow to give a complete geometric description of \((C,p)\) when \(\deg C =3,4\) and to describe some subgroups of Aut(\(C\)). Notice that to every pair \((C,p)\) we can associate a surface fibred in hyperelliptic curves which are the double cover of lines \(L\) through \(p\), ramified on \(L\cap C\) or on \(L\cap (C-\{p\})\). The pair \((C,p)\) has constant moduli if and only if the family of hyperelliptic curves is locally trivial.