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- E. Shustin
- 2002

We prove new patchworking theorems for singular algebraic curves, which state the following. Given a complex toric threefold Y which is fibred over C with a reduced reducible zero fiber Y 0 and other fibers Y t smooth, and given a curve C 0 ⊂ Y 0 , the theorems provide sufficient conditions for the existence of one-parametric family of curves C t ⊂ Y t ,… (More)

- E. SHUSTIN
- 2006

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… (More)

- Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin
- 2009

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.

- Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin
- 2008

To the memory of Andrey Bolibruch, a lively man of creative mind and open soul Abstract The Welschinger numbers, a kind of a real analog of the Gromov-Witten numbers which count the complex rational curves through a given generic collection of points, bound from below the number of real rational curves for any real generic collection of points. By the… (More)

- E. Shustin
- 2006

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… (More)

- E. Shustin
- 2006

The Welschinger invariants of real rational algebraic surfaces are natural analogues of the Gromov-Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces, equipped with a… (More)

- Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin
- 2003

Welschinger's invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using Mikhalkin's approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer d, there exists a real… (More)

Let L be a set of n lines in R d , for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [8],… (More)

- EUGENII SHUSTIN
- 2007

We suggest a version of Nullstellensatz over the tropical semiring, the real numbers equipped with operations of maximum and summation.