Variational formulae for Fuchsian groups over families of algebraic curves (Q5933521)

The author studies the problem of understanding the uniformizing Fuchsian groups for a family of plane algebraic curves by determining explicit first variational formulae for the generators. More explicitely, the data are a compact Riemann surface \(X_0\) of genus \(g>1\) with a given plane algebraic model \(X_0=\{(x,y) \in\mathbb{C}^2: P(x,y)=0\}\), where \(P(Z,Y)= \sum_{i,j}a_{ij} X^iY^j\in \mathbb{C} [X,Y]\), the group \(G_0\) uniformizing it, i.e., \(X_0=U/G_0\), where \(U\) is the upper half-plane, and a family of polynomials \(P_t(X,Y)= \sum_{i,j}a_{ij} (t) X^iY^j\in \mathbb{C}\{t\} [X,Y]\), where \(a_{ij}(0)=0\). For each \(T\) in a small disk around the origin, the author determines first variational formulae for the elements in the group \(G_t\) uniformizing the compact Riemann surface \(X_t= \{(x,y) \in\mathbb{C}^2: P_t(x,y)=0\}\).