Systems of polynomial equations defining hyperelliptic \(d\)-osculating covers
Let \(X\) denote a fixed smooth projective curve of genus \(1\) defined over an algebraically closed field \(\mathbb{K}\) of arbitrary characteristic \(p\neq 2\). The author assumes the degree-2 projection \(\varphi: X \to {\mathbb P}^1\) ramified over \(\{\infty, 0, 1, \lambda\}\); \(X\) is expressed by \(y^2 = x(x-1)(x-\lambda)\) in the ordinary sense. As a typical case, by letting the affine coordinate \(x = P/Q\) where \(P\) and \(Q\) are certain polynomials over \({\mathbb P}^1\), there naturally appears a covering space of \(X\). Based on the covering space, the author investigates the moduli space \(H(X,n)\) of degree-\(n\) finite separable covers of \(X\) by hyperelliptic curves. It is related to a square system of polynomial equations. Furthermore, the author obtains nonsquare systems parameterizing hyperelliptic \(d\)-osculating covers for any \(d>1\).
scientific article; zbMATH DE number 6485784
scientific article