## Gluing and ungluing curves of low genus

From the problem of determining the endomorphism algebra of an abelian variety – typically a Jacobian – one is naturally led to consider an operation of *gluing* between curves of low genus: given curves C1, C2 of genera g1, g2 respectively, if g1+g2 < 4 there are countably many ways to glue C1 and C2 to get a new curve C of genus g1+g2. This gluing operation is easy to describe at the level of the Jacobians of C1, C2 and C, but it would be interesting to reformulate it purely in terms of the geometry of the curves.