Degré des diviseurs sur les familles de courbes de \({\mathbb{P}}^ 3\). (Degree of divisors on families of curves of \({\mathbb{P}}^ 3)\) (Q1087937)

Let \(U_{d,g}\to H_{d,g}\) be the universal family of connected smooth curves of \({\mathbb{P}}^ 3\) of genus \(g\) and degree \(d.\) The relative tangent bundle and the hyperplane divisor of \({\mathbb{P}}^ 3\) define on \(U_{d,g}^ a \)divisor whose (relative) degree is \(\delta =g.c.d.(2g- 2,d)\). Assume \(d\geq \frac{5}{4}g+1\), so that \(H_{d,g}\) is connected. The author proves that if d is large enough with respect to g, then the degree of every divisor on \(U_{d,g}\) is a multiple of \(\delta\). This result provides an analog for \({\mathbb{P}}^ 3\) to a theorem of Enriques and Chisini stating, in modern terms, that on the universal family of curves of \({\mathbb{P}}^{g-1}\) of genus \(g\geq 3\) and degree \(2g-2\) every divisor has degree multiple of 2g-2.