Monodromy of closed points on generic hypersurfaces (Q2240450)

Let \(K\) be a field with algebraic closure \(\bar{K}\), \(L\subset K\) a finite separable extension of \(K\) and \(N\) the normal closure of \(L\) in \(K\). Then \(\mathrm{Gal}(N/K)\) is called the monodromy group of \(L/K\). The author shows that on the generic degree-\(d\) hypersurface in \(\mathbb{P}^{n+1}\), when \(d\geq n+2\), every closed point of degree \(d\) is separable and has monodromy group \(S_d\). He studies related Franchetta-type questions on generic objects and gives affirmative answers in specific cases. The Franchetta theorem asserts that every line bundle on the generic curve is a power of the canonical bundle.