Tropical Intersections: Where They Go Wrong, and Where They Go Right

Tropical geometry is a powerful tool for understanding curves and other varieties. Often described as a skeletonized version of algebraic geometry, it reduces potentially ineffable objects to piecewise-linear ones. These combinatorial objects are often much easier to analyze than the originals, and can be used to piece together information about the schemes from whence they came. Understanding precisely when information can be lifted is therefore of the utmost importance. A question in this vein is when tropicalization commutes with intersection. For two curves (or varieties, or schemes, depending on your preference) X and X ′, we know that Trop(X ∩ X ′) ⊂ Trop(X) ∩ Trop(X ′); that is, actual intersection points are still intersection points in the tropicalization. However, the opposite direction of containment does not always hold: tropical curves may have intersection points not corresponding to points of the original curves. After outlining when intersections are nice (essentially when codimensions are nice), we will delve into some not-so-nice intersections, which can include components of positive dimension and even noncompact components. Fortunately, the work of [OR] allows us to make some sense out of these “bad” cases, by use of judicious compactification and multiplicity counting. So perhaps a more accurate but less catchy title for this project would have been Tropical Curves: Where They Go Right, and Where They Go Wrong, and Where They Go Kind Of OK After All. After outlining these results and methods, we offer examples illustrating various cases of intersections, and explore the freedom and restrictions we have for where “actual intersections” map to in the tropical intersections. We will first establish our notation and conventions in accordance with [OR], and then mention the specific conditions we’ll usually be keeping in mind throughout the paper. Let K be a non-Archimedean field with valuation val : K → R ∪ {∞}, and take K to be complete or algebraically closed. Let T ∼= Gm be a finite-rank split torus over K with coordinate functions x1, ..., xn. We define the tropicalization map from the closed points of T to R by trop : |T| → R, trop(ξ) = (val(x1(ξ)), ..., val(xn(ξ))). For a closed subscheme X ⊂ T, the tropicalization of X, written trop(X), is the (Euclidean) closure of the set trop(|X|) in R. For our intents and purposes, we will usually take K to be the field of Puiseux series over C with uniformizer t. Recall that this is the set of all sums ∑ s∈S⊂Q ckt , where S has a least element and there is a bound on the denominators of the elements of S, and where ck ∈ C. This field is algebraically closed, and has val(0) =∞ and val (∑ s∈S⊂Q ckt s ) = min{s |s ∈ S}