Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry
By Eugenii Shustin. 2002
52 citations recorded by the source corpus.
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.
View Paper: https://www.semanticscholar.org/paper/7fde7bef33cbff3b1b5c25d3a36669b0bd01ba04
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
- Thorsten Theobald
- Bernd Sturmfels
- Jürgen Richter-Gebert
- G. Mikhalkin
- Marianne Akian
- Eric Katz
- Federico Ardila
- Stephane Gaubert
Citations in the corpus
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