Notes on Tropical Geometry for the Gross-Siebert Program Seminar
These notes accompany a talk I gave at the virtual Gross-Siebert Program Seminar, organized by Suraj Dash.
Conventions and Reference List
Tropical Geometry and Mirror Symmetry by Mark Gross. Chapter 1 of this book is the main reference.
"First Steps in Tropical Geometry" by Richter-Gerbert, Sturmfels, Theobald. Gross says chapter 1 of his book mostly follows this reference. I looked at it a few times while prepping these notes.
Introduction to Tropical Geometry by Maclagan and Sturmfels. Used to cross-reference the more opaque definitions in Gross's book. This is a little more down-to-earth and typically provides enlightening examples.
Note that we use \(M \cong \mathbb Z^n\) and \(N \cong \operatorname{Hom}(M, \mathbb{Z})\). This is the opposite of the convention I'm used to, but it's what Gross uses in his book. As always \(M_\mathbb{R} = M\otimes_\mathbb{Z} \mathbb{R}\) and \(N_\mathbb{R} = \operatorname{Hom}_\mathbb{Z}(M,\mathbb R)\).
Parameterized Tropical Curves
We skip over the basics of tropical polynomials, tropical curves in \(\mathbb{R}^n\) and fans and go straight to section 1.3 of Gross's book, where we first encounter generalized tropical curves built from the data of weighted graphs and marked graphs. I gave a (bad) introduction to cones and fans here, but the unfamiliar reader should either check out section 1.2 of Gross's book or look at Fulton's Introduction to Toric Varieties. Note that any tropical curve \(V(f)\) locally looks like a fan around every vertex after translating to the origin.
Balancing condition
We first need to discuss the balancing condition. Let \(\Sigma \subset \mathbb{R}^n\) be a one-dimensional rational fan; i.e. a fan comprised only of the origin and a collection of \(s\) rays. Let \(v_i\) be the first lattice point of the \(i\)th ray in \(\Sigma\). We give \(\Sigma\) the structure of a weighted fan by assigning a positive integer \(m_i \in \mathbb Z\) to \(v_i\), and we say that \(\Sigma\) is balanced if
\[ \sum m_i v_i = 0.\]Here's an example from Maclagan and Sturmfels page 111:
Before we generalize this, let's review some terminology.
A polyhedron \(P\subset \mathbb{R}^n\) is the intersection of finitely many closed half-spaces.
A face of a polyhedron \(P\) is determined by a linear functional \(\lambda \in \mathbb (\mathbb{R}^n)^\vee\) like so. (Note that the codimension 1 faces of \(P\) occur on the of a single supporting hyperplane, the codimension 2 faces occur on the intersection of two supporting hyperplanes, etc.)
A polyhedral complex \(\Sigma\) is a collection of polyhedra such that
if \(P\in \Sigma\) then every face of \(P\) is also in \(\Sigma\) and
if \(P, Q\in \Sigma\) then \(P\cap Q \in \Sigma\).
The polyhedra of \(\Sigma\) are called cells. Cells which aren't faces of any larger cell are called facets. Notice that all the facts need not have the same dimension, which is why...
...we say that \(\Sigma\) is pure of dimension \(d\) if every facet of \(\Sigma\) has the same dimension.
The affine span of a polyhedron \(P\) is the smallest affine subset of \(\mathbb R^n\) which contains \(P\). The relative interior of \(P\) is the interior of \(P\) inside its affine span.
There are two more definitions we need, included here for reference.
Definition 1: Normal Fan Let \(P\subset \mathbb R^n\) be a polyhedron. The normal fan of \(P\) is the polyhedral fan \(\mathcal N_P\) consisting of the cones
\[\mathcal N_P(F) = \operatorname{cl}(\{w \in \mathbb (R^n)^\vee ~\mid~ \operatorname{face}_w(P) = F\}),\]as \(F\) varies over the faces of \(P\).
Definition 2: Star of a Cell Let \(\Sigma\) be a polyhedral complex in \(\mathbb R^n\) and let \(\sigma\) be a cell in \(\Sigma\). The star of \(\sigma \in \Sigma\) is a fan in \(\mathbb R^n\) denoted \(\operatorname{star}_\Sigma(\sigma)\). Its cones are indexed by those cells \(\tau \in \Sigma\) which contain \(\sigma\) as a face. The cone of \(\operatorname{star}_\Sigma(\sigma)\) which is indexed by \(\tau\) is the subset
\[ \overline{\tau} = \{\lambda(x - y) ~\mid~ \lambda\geq 0, x\in \tau, y\in \sigma\}.\]Definition 2 above is taken directly from Maclagan and Sturmfels. Note that they don't ask that polyhedral fans contain a zero-dimensional cone, so it's fine that \(\overline{\sigma}\) is the minimal dimensional cone of \(\operatorname{star}_\Sigma(\sigma)\).
We can now extend the balancing condition to arbitrary fans.
Definition 3: Balancing Condition Let \(\Sigma\) be a rational fan in \(\mathbb R^n\), pure of dimension \(d\), and suppose we have a weighting function \(m\) which assigns weights \(m(\sigma) \in \mathbb N\) for all the maximal cones \(\sigma\). Given \(\tau \in \Sigma\) of dimension \(d-1\), let \(L\) be the linear space parallel to \(\tau\). The abelian group \(L_\mathbb{Z} = L\cap \mathbb Z^n\) is free of rank \(d-1\) since \(\tau\) is a rational cone, and then \(N(\tau) = \mathbb Z^n/L_\mathbb Z \cong \mathbb Z^{n-d+1}\).
Now, for each maximal cone \(\sigma\) with \(\tau \subsetneq \sigma\), the set \((\sigma +L)/L\) is a one-dimensional cone in \(N(\tau)\otimes_\mathbb{Z} \mathbb R\) – it's just the projection of \(\sigma\) onto \(L\). Take \(v_\sigma\) to be the first lattice point on this ray. Then \(\Sigma\) is balanced at \(\tau\) if
\[\sum m(\sigma) v_\sigma = 0,\]iterating over all \(\sigma\) containing \(\tau\). We say that the fan \(\Sigma\) is balanced if it's balanced at all codimension (\(d-1\))-cones.
If we instead have a rational polyhedral complex \(\Sigma\) of pure dimension \(d\) with weights \(m(\sigma)\in \mathbb N\) on all maximal cells, then for each \(\tau \in \Sigma\) the fan \(\operatorname{star}_{\Sigma}(\tau)\) inherits the weighting function \(m\). We therefore say that \(\Sigma\) is balanced if \(\operatorname{star}_{\Sigma}(\tau)\) is balanced for all \((d-1)\)-dimensional cells \(\tau\).
Tropical curves from graphs
We can use the balancing condition to define tropical curves in a more abstract setting than in \(\mathbb R^n\). Let \(\overline{\Gamma}\) be a connected graph with no bivalent vertices (i.e. no vertices with exactly two connected edges). We denote by \(\overline{\Gamma}^{[0]}\) and \(\overline{\Gamma}^{[1]}\) the set of vertices and edges respectively of \(\overline{\Gamma}\). It's convenient to freely switch back and forth between thinking of \(\overline{\Gamma}\) as the purely combinatorial object defined by \(\overline{\Gamma}^{[0]}\) and \(\overline{\Gamma}^{[1]}\) and as a topological space given a union of closed line segments.
Denote by \(\overline{\Gamma}^{[0]}_\infty\) the set of univalent vertices of \(\overline{\Gamma}\), and write
\[\Gamma = \overline{\Gamma} \setminus \overline{\Gamma}^{[0]}_\infty.\]Thinking of \(\Gamma\) and \(\overline{\Gamma}\) as topological spaces, we see that \(\Gamma\) is a "graph with some non-compact edges", i.e. "legs" or "half-edges" and \(\overline{\Gamma}\) is its closure. This explains the notation, especially if we consider the univalent vertices of \(\overline{\Gamma}\) to occur at infinity.
Here are some rapid-fire definitions.
We let \(\Gamma^{[1]}_\infty\) denote the non-compact edges of \(\Gamma\).
A flag of \(\Gamma\) is a vertex/edge pair \((V,E)\in\Gamma^{[0]}\times \Gamma^{[1]}\) with \(V \in E\).
A weight function is a map assigning positive integer weights to the edges of \(\Gamma\):
A marked graph is a tuple \((\Gamma, x_1, ..., x_k)\) where \(\Gamma\) is as above and \(x_1,...,x_k\) are labels assigned to the non-compact edges with weight \(0\), i.e. \(\{E_{x_1},...,E_{x_k}\} \subseteq \Gamma^{[1]}_\infty\) is precisely the set of non-compact edges such that \(w(E_{x_i}) = 0\).
And finally the actual definition to which we've been building.
Definition 4: Parameterized Tropical Curve A marked parameterized tropical curve \[ h:(\Gamma, x_1,...,x_k)\to M_\mathbb{R} \] is a continuous map \(h\) satisfying the following two properties:
If \(E\in \Gamma^{[1]}\) and \(w(E) = 0\), then \(h|_E\) is constant; otherwise, \(h|_E\) is a proper embedding of \(E\) into a line of rational slope in \(M_\mathbb R\).
The balancing condition. Let \(V\in \Gamma^{[0]}\) and let \(E_1,...,E_\ell\in \Gamma^{[1]}_\infty\) be the edges adjacent to \(V\). Let \(m_i \in M\) be a primitive tangent vector to \(h(E_i)\) pointing away from \(h(V)\). Then
\[ \sum_{i = 1}^\ell w(E_i) m_i = 0. \] If \(h:(\Gamma, x_1,...,x_n)\to M_\mathbb{R}\) is a marked parameterized tropical curve, we write \(h(x_i)\) for \(h(E_{x_i})\). The genus of \(h\) is \(b_1(\Gamma)\).
Here are some pictures justifying the idea that as long as \(\overline{\Gamma}\) is not bivalent and the balancing condition is satisfied, then \(\operatorname{img}(h)\) does indeed look like a tropical curve.
We call these parameterized tropical curves because the image \(\operatorname{img}(h)\) of \(h\) is a tropical curve in \(M_\mathbb R\) and \(h\) parameterizes each "edge" of the tropical curve according to condition (1) in the definition.
We say that two marked parameterized tropical curves \(h:(\Gamma,x_1,...,x_k)\to M_\mathbb{R}\) and \(h':(\Gamma',x_1',...,x_k')\to M_\mathbb R\) are equivalent if there is a homeomorphism \(\varphi:\Gamma \to \Gamma'\) with \(\varphi(E_{x_i}) = E_{x_i'}\) and \(h = h'\circ \varphi\). We define a marked tropical curve to be an equivalence class of parameterized marked tropical curves.