# Connected sum of algebraic curves, handlebody decomposition, and induction on genus

Are there any nice methods of taking algebraic curves $$C_1, C_2$$ of genera $$g_1,g_2$$ and producing a curve $$C_3 = C_1 \# C_2$$ of genus $$g_3 = g_1+g_2$$? I'm imagining doing this over any field, but taking intuition from the complex case.

To be "nice", one would like to produce from two smooth (connected) curves another smooth (connected) curves, and presumably be able to use the construction for inductive proofs on genus.

Gluing two curves at a point will violate smoothness, and the resolution of the singularity will just separate the two curves, so this approach won't work. Also, the local ring at a point is a birational invariant, which again seems to doom that approach.

Instead of blowing up to resolve the singularity, one could try to situate the gluing as a degeneration within a generically smooth family of curves. This seems more non-unique than I was initially hoping for, but is this the best approach? Are there some really nice examples of inductive proofs with that technique? What properties would one expect a generic member to share with the (two components of the) singular curve, and what properties would differ? One application I may find particularly intriguing is using rational points on smaller genus curves to get points on larger genus curves, so families in which a single point stays rational would be interesting.

One could reverse the line of thought to form "handlebody decompositions" of a given smooth curve. Is this always possible? What about a total decomposition into many curves of genus 1 (or 0)? Are there alternative ways of forming such a decomposition?

Degeneration and deformation techniques are very common, so I'm sure to the extent any of this is useful, it's old hat to algebraic geometers, but I would appreciate someone pointing in a direction that would satisfy my curiosity.

First, glue two points, obtaining a singular curve of genus $$g_1+g_2$$. Then smooth out the singularity. Equivalently, remove two discs, one from each curves, and glue two small annuli around the discs using the relations $$xy=t$$, where $$x$$ and $$y$$ are local coordinates on the curves and $$t$$ is a non-zero fixed small complex number.