For any $k$ dimensional algebraic variety $X$ in the complex torus $(\Bbb C^*)^n$ there is a good compactification, i.e. a compact smooth toric variety $M\supset (\Bbb C^*)^n$ such that the closure $\overline X$ of $X$ in $M$ does not intersect orbits of $M$ whose dimension is smaller than $n-k$. Let $A$ be a positive 0-cycle (a finite set of points equipped with positive integral multiplicities) belonging to the union $M^{n-k}$ of $(n-k)$-dimensional orbits of $M$.

{\bf Problem} {\it Find all $Y\subset M$ with $\dim Y=k$ such that $Y$ does not intersect orbits of $M$ whose dimension is smaller that $n-k$ and the intersection of $Y$ with $M^{n-k}$ is the 0-cycle $A$}.

About 25 years ago I solved this problem [1] for the case when $n=2$. It turns out that in this case for existence of a solution $Y$ the 0-cycle $A$ has to satisfy {\it the additive and the multiplicative balance conditions} and {\it the rigidity conditions}. If these conditions are satisfied then one can explicitly describe all solutions $Y$. Only recently I discovered a complete solution for the case when $\dim Y=n-1$ (not published yet). One can easily prove that the additive balance conditions have to be satisfied in general case. These conditions play a key role in tropical geometry. In the talk I will explain why the multiplicative balance conditions and rigidity conditions have to be satisfied in general case. I plan to enlarge tropical geometry using these results.

References

1. A.~Khovanskii. Newton polygons, curves on torus surfaces, and the converse Weil theorem, Russian Math. Surveys 52 (1997), no. 6, 1251--1279.