Download Algorithmic aspects of tropical intersection theory

2.5. Intersection products in Rn
so assume n − d ≥ 3. Let S be any subdivision class of X (Note that, while S depends
on the particular polyhedral structure of X, T does not). We now pick a vector w ∈ Rn
with the following properties (we write VS ∶= ⟨S⟩R for a subdivision class S):
• w ∉ VS
• For any subdivision class S ′ with dim(VS ′ ∩ VS ) ∈ {d − 1, d − 2}, we have that
w ∉ VS + VS ′ . Note that dim(VS + VS ′ ) = 2d − dim(VS ′ ∩ VS ) ≤ n − 1.
Since we only exclude a union of linear spaces of dimension at most n − 1, such a w
exists. We now consider the projection πw ∶ Rn → Rn / ⟨w⟩ ≅ Rn−1 . The push-forward
πw∗ (X) is still a variety of dimension d, so the codimension has decreased by one. Pick
a suitable polyhedral structure on X compatible with the map πw (again, this does
not change the support of the subdivision classes). By our choice of w, πw∣S is injective
and πw (S) is still a subdivision class. Hence it must be convex by induction.
Example 2.4.11. The natural question to ask next is of course when the subdivision
classes form a polyhedral complex. A necessary condition is connectedness in codimension one: Consider two linear planes in R4 intersecting in a point. If we equip
both planes with arbitrary weights, we can refine them such that we obtain a tropical
variety X. However, the subdivision support of X obviously consists of the two planes,
which do not intersect in a common face.
Conjecture 2.4.12. Let X be a tropical variety and assume X is locally connected
in codimension one. Then the subdivision supports of X form a polyhedral complex,
which is the coarsest polyhedral structure on X.
Remark 2.4.13. Note that we have stated these results and conjectures only for
tropical varieties, i.e. assuming that all weights are positive. Of course the proof of
Proposition 2.4.10 works equally well for arbitrary cycles. However, for general weights
it is already unclear in codimension one how the subdivision classes behave. Hence we
prefer to restrict ourselves to varieties for now.
2.5. Intersection products in Rn
There are two main equivalent definitions for a tropical intersection product in Rn ,
the fan displacement rule [RGST] and via rational functions [AR2]. At first sight,
the computationally most feasible one seems to be the latter, since we can already
compute it with the means available to us so far:
Let X, Y be tropical cycles in Rn and ψi = max{xi , yi } ∶ Rn × Rn → R. Denote by
π ∶ Rn × Rn → Rn the projection onto the first n coordinates. Then we define
X ⋅ Y ∶= π∗ (ψ1 ⋅ ⋅ ⋅ ⋅ ⋅ ψn ⋅ (X × Y ))
(Here, applying π∗ just means forgetting the last n coordinates) However, computing
this directly turns out to be rather inefficient. The main reason is that, since we
compute on the product X × Y , we multiply the number of their maximal cones by
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