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The patchworking construction in tropical enumerative geometry. (English) Zbl 1105.14036
Lossen, Christoph (ed.) et al., Singularities and computer algebra. Selected papers of the conference, Kaiserslautern, Germany, October 18–20, 2004 on the occasion of Gert-Martin Greuel’s 60th birthday. Cambridge: Cambridge University Press (ISBN 0-521-68309-2/pbk). London Mathematical Society Lecture Note Series 324, 273-300 (2006).
In the early 1990’s the author of this article suggested to use the patchworking construction for tracing properties of objects, defined by polynomials, other than in the original method invented by O. Viro in 1979-80. In the paper under review the author restricts himself to families of surfaces, and curves in these surfaces, and he traces the property to possess a certain collection of singularities. As a result he states in section 2.3 a patchworking theorem for curves on toric surfaces which is proved in section 2.4. This theorem is sufficient for applications to a tropical calculation of Gromov-Witten and Welschinger invariants on toric surfaces.
The general situation in the patchworking construction is a one-parametric flat family of surfaces with irreducible fibers \(Y_t\), \(t\neq 0\), and reduced central fiber \(Y_0\) that contains the zero locus \(X_0\) of a section of the restriction to \(Y_0\) of a line bundle on \(Y\). In most previous papers it is assumed that the components of \(X_0\) are reduced and they meet transversally the singular locus of \(Y_0\). A major novelty of the work under review is that \(X_0\) is allowed to have nonreduced components that may be not transversal to \(\text{ Sing}(Y_0)\). This covers the degenerations which appear in the tropical enumeration of curves.
For the entire collection see [Zbl 1086.14001].

14H15 Families, moduli of curves (analytic)
14H20 Singularities of curves, local rings
14P05 Real algebraic sets
14M25 Toric varieties, Newton polyhedra, Okounkov bodies