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Abstract. A detailed algebraic-geometric background is presented for the tropical approach to enumeration of singular curves on toric surfaces, which consists of reducing the enumeration of algebraic curves to that of non-Archimedean amoebas, the images of algebraic curves by a real-valued non-Archimedean valuation. This idea was proposed by Kontsevich and recently realized by Mikhalkin, who enumerated the nodal curves on toric surfaces. The main technical tools are a refined tropicalization of one-parametric equisingular families of curves and the patchworking construction of singular algebraic curves. The case of curves with a cusp and the case of real nodal curves are also treated. §1. IntroductionThe rapid development of tropical algebraic geometry over recent years has led to interesting applications to enumerative geometry of singular algebraic curves, proposed by Kontsevich [16]. The first result in this direction was obtained by Mikhalkin [18], who counted the curves with a given number of nodes on toric surfaces via lattice paths in convex lattice polygons. Our main goal in the present paper is to explain this breakthrough result, notably the link between algebraic curves and non-Archimedean amoebas, which is the core of the tropical approach to enumerative geometry. Our point of view is purely algebraic-geometric and differs from Mikhalkin's method, which is based on symplectic geometry techniques. Briefly speaking, we count equisingular families of curves over a punctured disk. The tropicalization procedure extends such families to the central point, and these tropical limits are basically encoded by non-Archimedean amoebas. In its turn, the patchworking construction restores an equisingular family out of the central fiber.Tropicalization. Let ∆ ⊂ R 2 be a convex lattice polygon, and let Tor K (∆) be the toric surface associated with the polygon ∆ and defined over an algebraically closed field K of characteristic zero. We denote by Λ K (∆) the tautological linear system on Tor K (∆) generated by the monomials x i y j , (i, j) ∈ ∆ ∩ Z 2 . We would like to count the n-nodal curves belonging to Λ K (∆) and passing through r = dim Λ K (∆) − n = |∆ ∩ Z 2 | − 1 − n generic points in Tor K (∆), i.e., we want to find the degree of the so-called Severi variety Σ ∆ (nA 1 ). Let K be the field of convergent Puiseux series over C, i.e., power series of the form b(t) = τ ∈R c τ t τ , where R ⊂ Q is contained in an arithmetic progression bounded from below, and τ ∈R |c τ |t τ < ∞ for sufficiently small positive t. The field K is equipped with a non-Archimedean valuation Val(b) = − min{τ ∈ R : c τ = 0}, which A curve C ∈ Λ K (∆) with n nodes is given by a polynomialWithout loss of generality we can assume that all exponents of t in a ij (t) are integral, and thus, polynomial (1.0.1) determines an analytic surfacewhere D is a small disk in C centered at 0, and X is such that the fibers X t are complex algebraic curves that belong to the linear system Λ(∆) on the surface Tor(∆) and have n nodes (cf. Lemma 2.3 and Sub...

We define a series of relative tropical Welschinger-type invariants of real toric surfaces. In the Del Pezzo case, these invariants can be seen as real tropical analogs of relative Gromov-Witten invariants, and are subject to a recursive formula. As application we obtain new formulas for Welschinger invariants of real toric Del Pezzo surfaces.

Abstract. The Welschinger invariants of real rational algebraic surfaces are natural analogues of the genus zero Gromov-Witten invariants. We establish a tropical formula to calculate the Welschinger invariants of real toric Del Pezzo surfaces for any conjugation-invariant configuration of points. The formula expresses the Welschinger invariants via the total multiplicity of certain tropical curves (non-Archimedean amoebas) passing through generic configurations of points, and then via the total multiplicity of some lattice paths in the convex lattice polygon associated with a given surface. We also present the results of computation of Welschinger invariants, obtained jointly with I. Itenberg and V. Kharlamov.

Abstract. We prove that there exists a positive α such that for any integer d ≥ 3 and any topological types S 1 , . . . , Sn of plane curve singularities, satisfying µ(S 1 ) + · · · + µ(Sn) ≤ αd 2 , there exists a reduced irreducible plane curve of degree d with exactly n singular points of types S 1 , . . . , Sn, respectively. This estimate is optimal with respect to the exponent of d. In particular, we prove that for any topological type S there exists an irreducible polynomial of degree d ≤ 14 µ(S) having a singular point of type S.