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One of the motivations to study tropical geometry is that there are some hard Algebraic Questions that can be answered by proving them in the Tropical World. For example one can show that tropical Bezout's Theorem implies the Algebraic Bezout.

What properties are there known that are true (or might be) in tropical geometry that don't imply that their algebraic version is true?

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The topology of an algebraic variety is not determined by the topology of the tropicalization. For example, the tropicalization of an elliptic curve has first betti number zero. –  J.C. Ottem Aug 26 '11 at 13:14
    
Maybe I am confused, but I think that the tropicalization of an elliptic curve has typically (generically?) first betti number one. –  Bruno Martelli Aug 30 '11 at 13:32
    
Yes, the Betti number of the tropicalization will be zero or one depending on the sign of the valuation of the j-invariant. –  J.C. Ottem Sep 10 '11 at 14:21
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up vote 6 down vote accepted

There is a simple nice fact which holds in the tropical plane that has no counterpart in algebraic geometry (nor in any kind of standard geometry I might think of): two tropical lines always "intersect" in a single point... even if they coincide!

Of course this property relies on the fact that "intersection" is not defined in the usual way. We define the "intersection" of two tropical curves $C_1$ and $C_2$ as follows: the union $C_1\cup C_2$ has a natural cellularization into vertices and edges, and "$C_1\cap C_2$" is the union of the vertices contained in the set-theoretic intersection $C_1\cap C_2$. One may also define a multiplicity on each intersection point. With this definition, the intersection of a tropical curve with itself is the union of its vertices.

Therefore two (possibly coinciding) tropical lines always intersect in a point. By defining anagously an appropriate (dual) notion of "span", the dual sentence is also true: two (possibly coinciding) points always span a single line.

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Isn't this the definition of a sort of "generic intersection" or "intersection up to deformation", rather than of a genuine intersection? It looks like the concept is missing, while the property does have an algebraic counterpart. On the other hand, the dual statement is a bit more stunning; but, again, the trick could lie in the word "appropriate". –  Marco Golla Aug 25 '11 at 16:04
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I don't know if there is a generally accepted notion of "intersection" in tropical geometry, but I like very much this one. Even if you take it as a notion of "generic intersection", it has no counterpart in other geometries: no "generic intersection" is defined between a line and itself (usually, if you perturb the line, the intersection point can be any point on the line, so there is no convergence: in the tropical world however there is convergence) –  Bruno Martelli Aug 25 '11 at 16:21
    
@BM: what do you mean by "convergence"? Isn't the phenomenon of two lines in the plane intersecting in one point analogous to what happens with the intersection product in the Chow ring of -say- $\mathbb{P}^2$ ? –  Qfwfq Aug 25 '11 at 16:42
    
By "convergence" I mean that you can take an alternative definition of the "intersection" "$C_1 \cap C_2$" by taking the limit (as $t\to 0$) of the set-theoretic intersection $C_1 \cap C_2^t$, where $C_2^t$ is obtained by translating $C_2$ at distance $t$ on a generic direction $v$ (the result does not depend on $v$). The Chow ring seems different to me, since there in order to have a well-defined "intersection" you first need to quotient all curves that are rationally equivalent (in particular, there is only one line in the Chow ring of $P^2$) –  Bruno Martelli Aug 25 '11 at 17:07
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