Tropical Geometry and Affine Buildings

Jenia Tevelev (with Sean Keel)

July 14, 2007

1 Tropical algebraic geometry

Consider the pictures in Figure 1. The objects are roughly the same in dimension 1 (see [1]). We will discuss the case of dimension 2 and higher in this lecture. This is joint work with Keel [2].

Let X be an algebraic variety. There are two main ways in which the tropicalization T (X) arises—as the degeneration of X or as the compactification of X.

Suppose X is over ℂ, and let K = ℂ((z)) be the field of Puiseux series in z with coefficients in ℂ. Then X(K) degenerates to a compactification \overline{X}(ℂ) under γ(z) ↦ γ(0). On the other hand, there is a map from the set of valuations of the function field k(X) to \overline{X} sending a valuation v to the center of v, which is a scheme theoretic point in \overline{X}.

Let O^{∗}(X) be the invertible rational functions on X and k the function field of X. Then M = O^{∗}(X)/k^{∗} is an integer lattice, isomorphic to some ℤ^{r}. The tropical variety T (X) lives in Hom(M, ℚ).

Definition (Non-archimedean amoeba definition). The tropical variety T (X) is the set {[γ] : γ ∈ X(K)}, where [γ](m) := deg[m(γ(z))] for m ∈ O^{∗}(X).

Definition (Divisorial definition). The tropical variety T (X) is the set

{[v] : v is a divisorial valuation of ℂ(X)},

where [v](m) := v(m) for m ∈ O^{∗}(X).

Suppose X ⊂ \overline{X} is such that \overline{X} is smooth, \overline{X}\X has normal crossings, and \overline{X}\X = D_{1} ∪ ⋯ ∪ D_{s} are divisors, where [D_{i}] = [val_{D_{i}}] ∈ T (X). Then

T (X) = ⋃_{D_{i_{1}}∩···∩D_{is}≠∅} ℚ_{≥0}[D_{i_{1}}] + ⋯ + ℚ_{≥0}[D_{i_{s}}].

Figure 1: A Bruhat-Tits building, a tropical line, a semistable rational curve, and a stable rational curve.

Examples.

1. Let X = ℙ^{1}\\{a_{1}, …, a_{n−1}, ∞\}. Then \overline{X} = ℙ^{1}, and O^{∗}(X)/ℂ^{∗} = \{x−a_{1}, x−a_{2}, …, x− a_{n−1}}, and

[a_{1}] = (1, 0, …, 0), …, [a_{n−1}] = (0, 0, …, 1), [∞] = (−1, −1, …, −1).

The tropical variety consists of n rays in the irections e_{1}, e_{2}, …, e_{n−1}, −(e_{1} + ⋯ + e_{n}).

2. Let X = ℙ^{2}\{L_{1}, …, L_{n}}, where L_{1}, …, L_{n} are lines in general linear position. Then ℙ^{2} is a compactification of X with normal crossing divisor boundary. The tropical variety T (X) is a two-dimensional fan on the rays [L_{1}], …, [L_{n}], and every pair of rays spans a cone in T (X) because every two lines intersect in ℙ^{2}. Now suppose L_{1} ∩ L_{2} ∩ L_{3} = {p} ∈ ℙ^{2}. Then we can blow up ℙ^{2} at p to obtain a compactification of X with normal crossing divisor boundary. After blowing up, the lines L_{1}, L_{2}, and L_{3} no longer intersect, but each meets the exceptional divisor E. So we have to replace the three cones ℝ_{≥0}[L_{1}] + ℝ_{≥0}[L_{2}], ℝ_{≥0}[L_{1}] + ℝ_{≥0}[L_{3}], and ℝ_{≥0}[L_{2}] + ℝ_{≥0}[L_{3}] from above with three new cones of the form ℝ_{≥0}[E] + ℝ_{≥0}[L_{i}] for i = 1, 2, 3.

Let R = ℂ[[z]]. Then X_{R} is smooth and proper over Spec R, and the complement has normal crossings:

X_{K}  X_{R} Spec K  SpecR

Let M̃ = O^{∗}(X)/R^{∗}. For each boundary divisor D, compute [D] ∈ Ñ, define the fan the same way, and intersect with the fiber for the map Ñ → ℤ.

Example. Let X = ℙ^{1}\{0, 1, z, ∞}. Then O^{∗}(X)/R^{∗} is generated by x, x − 1, x − z, and z. The divisors 0, z, and the special fiber over 0 intersect, so we blow up to get an exceptional divisor D_{2} and the pullback D_{1} of the special fiber:

[0] = (1, 0, 0, 0),  [1] = (0, 1, 0, 0),  [z] = (0, 0, 1, 0),
[∞] = (−1, −1, −1, 0),  [D_{1}] = (0, 0, 0, 1),  [D_{2}] = (1, 0, 1, 1).

Now the question is: How do we compute normal crossing models? We can compute limits of lines.

ℙ_{K}^{2}  ℙ_{R}^{2} Spec K  Spec R

Embeddings ℙ_{K}^{2} ↪ ℙ_{R}^{2} are classified by the affine Grassmannian B = PGL_{3}(K)/PGL_{3}(R). An element [Λ] ∈ B is an equivalence class of a lattice Λ ∈ K^{3}, where two lattices Λ_{1}, Λ_{2} are equivalent if Λ_{1} = z^{a}Λ_{2} for some integer a. This gives an arrangement of lines ℙ_{R,Λ}^{2} in ℙ_{R}^{2}. Let C_{Λ} be the limiting configuration of lines.

Figure 2: Blowing up ℙ^{2}.

Figure 3: A tropical line.

Definition. A membrane [M] ⊂ B is the set of [Λ] ∈ B such that C_{Λ} contains a triangle of lines.

What does a membrane look like?

The compactified building \overline{B} ⊃ B is the set of all free R-modules (of any rank) in K^{3}, up to equivalence. Let f_{1}, …, f_{n} ∈ K^{3} such that the hyperplane L_{i} is the vanishing set of f_{i}. Then Rf_{1}, …, Rf_{n} are on the boundary of \overline{B}.

Lemma. The membrane M is the tropical convex hull of Rf_{1}, …, Rf_{n}. In fact, M is the union of apartments generated by linearly independent f_{i_{1}}, f_{i_{2}}, f_{i_{3}}.

Theorem. We have

ℙ_{K}^{2} ↪ ∏_{Λ∈M} ℙ_{R}^{2}.

Let S be the closure with divisors L_{1}, …, L_{m}, smooth with normal crossing boundary.

Theorem. The membrane is homeomorphic to the tropical 2-plane, and the homeomorphism is given by

Ψ : B ⟶ ℚ^{n}/(1, 1, …, 1)
Λ ⟼ (min {a_{1} : z^{a_{1}}f_{1} ∈ Λ}, …, min {a_{n} : z^{a_{n}}f_{n} ∈ Λ})

The same works in general for ℙ^{m} for any m.

Figure 4: Compactified affined building and compactification of X.

References

[1] Mikhail Kapranov: Veronese curves and Grothendieck-Knudsen moduli space M(0,n), Journal of Algebraic Geometry 2 (1993), 239–262

[2] Sean Keel and Jenia Tevelev: Geometry of Chow Quotients of Grassmannians, Duke Math. J. 134, no. 2 (2006), 259–311