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.

  • 1
    $\begingroup$ I am a bit confused. If you have a fan, then the set of divisors are given by the edges of this fan. And then from this you can compute the intersection pairing. $\endgroup$ Dec 12, 2017 at 17:08
  • $\begingroup$ @NicolasHemelsoet, thanks for your reply. Maybe I wasn't being precise. I know what the fan "looks" like. I know it has $n$ divisors (which I thought correspond to vertices, e.g. page 16 of arxiv.org/pdf/0706.1310.pdf). But beyond the visual appearance, I don't know anything more. In this article, the author describes an algorithm to compute the intersection numbers which seems to rely on a knowledge of the coordinates of the vertices. $\endgroup$ Dec 12, 2017 at 17:12
  • 1
    $\begingroup$ Ok I can try to make an answer, tell me if this is useful. $\endgroup$ Dec 12, 2017 at 17:16

1 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, $$ D \cdot C = \langle m_{\sigma} - m_{\sigma'},u \rangle $$

  • $\begingroup$ Thanks @Nicolas Hemelsoet! The toric variety corresponding to my situation is of the form $(\mathbb{C}^8 - F_{\Sigma})/(\mathbb{C}^\star)^5$. There are 8 divisors, and I know what the exclusion set $F_{\Sigma}$ is in terms of the 8 coordinates $(z_1, z_2, \ldots, z_8)$ - its just a finite union of codimension-2 spaces. So the variety is a 3-dimensional complex space. The Calabi-Yau condition then lets us focus on a surface. So in this case, does what you said also apply? Also, $a = (D_\rho)^2$ but I don't know what $D_\rho$ is. What does the notation $\mathbb{R}_+ v_1 = \rho$ mean? $\endgroup$ Dec 12, 2017 at 18:06
  • 1
    $\begingroup$ Hi, for your more specific situation I can try to think more (but I'm a bit in a rush right now). If $\rho$ is an edge of a fan then there is a $T$-invariant associated divisor $D_{\rho}$. $\rho = \Bbb R_+ v_1$ means that $v_1$ lies on the edge $\rho$. (I can try to provide more details and think to your specific situation later). $\endgroup$ Dec 12, 2017 at 18:13

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .