Recovering data about the toric fan from minimal information

Question

I am a physics grad student studying toric diagrams since they naturally arise in the description of certain gauge theories obtained by compactifying M/F-theory on Calabi-Yau 3-folds. In this context, the Calabi-Yau manifold is elliptically fibered, and the base of the fibration is a nonsingular two-dimensional toric variety.

This means -- as I understand -- that the toric fan describing the base is obtainable by succesive blowups of the Hirzebruch surface or $\mathbb{P}^2$.

In the physics context, the toric fan of the base is the dual diagram of the so called ``(p, q) web'' diagram. So, given a web diagram describing a field theory on the physics side, I can construct the dual diagram -- which is just the toric fan of the base.

However, in doing so, I do not have the information about the coordinates of the vertices (i.e. one-dimensional cones, which correspond to divisors) of the toric fan.

My question is, given such a diagram with the only additional information being that it describes the base of an elliptically fibered Calabi-Yau 3-fold, can I somehow reconstruct the "coordinates" of the vertices relative to any vertex as an origin? I thought there should be a simple way by experimenting with blow-ups of the basic two-dimensional toric surfaces, but none of those blowups seem to naturally reproduce the toric fan I am interested in.

So in more simple terms: given a toric fan (a diagram in two dimensions on a plane), how do I reconstruct the coordinates of its vertices (relative to some chosen coordinate frame)?

Are there any additional constraints that I am missing?

I need this information because I want to compute all the triple intersection numbers and one of the first steps is to derive linear equivalences between divisors, and compute the Mori cone generators.

Answer 1

This is really more a comment that an answer. For toric surface, the situation is pretty nice. Let me assume for simplicity that $X$ is smooth, that is for every cone $\sigma \in \Sigma$, $\sigma = \{v_1,v_2\}$ is smooth, meaning that $v_1,v_2$ generate the lattice $\Bbb Z^2$.

Now, we have an exact sequence $$ 0 \to \Bbb Z^2 \to \text{TDiv}(X) \to Pic(X) \to 0 $$

Where $Pic(X)$ is the Picard group of $X$, that is the group of divisor modulo linear equivalence. The first map simply send a character $\chi^m$ to its divisor $D = \text{div}(\chi(m))$.

Now, for computing self intersection of a divisor $D_{\rho}$ with primitive vector $v_1$ (this means that $\Bbb R_+ v_1 = \rho$ and $v_1 \in \Bbb Z^2$) corresponding to an edge is simple : such divisor is between two cones $\sigma = cone(u,v_1)$ and $\sigma' = cone(v_1,w)$. Then since by hypothesis the cone are smooth, $u + w = -a v_1$ for some $a \in \Bbb Z$, and we have $a = (D_{\rho})^2$. Other intersection numbers are fairly simple : $D_i \cdot D_j = 1$ if $i,j$ share a cone and $0$ else.

More informations can be found here, in particular there are notes about toric divisors, and later there is also toric Mori theory. The book by Cox, Little and Schenck is also an excellent reference.

There is a simple formula for the intersection of a $T$-invariant curve $C$ and a $T$-invariant divisor: this is proposition 6.3.8 in the book by Cox,Schenck and Little.

Proposition : Let $C$ a $T$-invariant curve corresponding to the wall $\tau = \sigma \cap \sigma'$. Let $D$ a Cartier divisor with Cartier data $m_{\sigma}$. Then, if $u \in \sigma'$ maps to a minimal generator of $N/\Bbb R \tau$. Then, $$ D \cdot C = \langle m_{\sigma} - m_{\sigma'},u \rangle $$