Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry

TL;DR

This paper introduces a new patchworking theorem for singular algebraic curves in complex toric threefolds, linking algebraic deformations with non-Archimedean amoebas and extending enumerative geometry methods.

Contribution

It provides a sufficient condition for constructing families of algebraic curves with prescribed singularities, connecting complex algebraic geometry with non-Archimedean amoebas and enumerative techniques.

Findings

Established a link between nodal curves over complex Puiseux series and non-Archimedean amoebas.

Extended patchworking techniques to curves with cusps and real nodal curves.

Connected algebraic deformations with tropical geometry methods.

Abstract

We prove a new patchworking theorem for singular algebraic curves, which states the following. Given a complex toric threefold $Y$ which fibers over ${\mathbb C}$ with a reduced reducible zero fiber $Y_0$ and other fibers $Y_t$ smooth, and given a reduced curve $C_0\subset Y_0$, the theorem provides a sufficient condition for the existence of a one-parametric family of curves $C_t\subset Y_t$, which induces an equisingular deformation for some singular points of $C_0$ and certain prescribed deformations for the other singularities. As application we give a comment on a recent theorem by G. Mikhalkin on enumeration of nodal curves on toric surfaces via non-Archimedean amoebas [arXiv:math.AG/0209253]. Namely, using our patchworking theorem, we establish link between nodal curves over the field of complex Puiseux series and their non-Archimedean amoebas, what has been done by Mikhalkin in a different way. We discuss also the case of curves with a cusp as well as real nodal curves.