# Smooth Fano Polytopes and Hypersurfaces

This is a rather extended question, so I will try to make it as compact and readable as possible. I am trying to practice with the Macaulay2 software, in particular the polyhedra and normaltoricvarieties packages. From the paper given here: http://arxiv.org/pdf/1209.3186v3.pdf, I have been using the polytope provided in Example 14, which is a smooth fano polytope in 4 dimensions. Inputting this as the matrix $$\begin{pmatrix} 1 & 0 & 0 & 0 & 1 & -1 & -1 & 0 & 1 & -1\\ 0 & 1 & 0 & 0 & -1 & 1 & 0 & -1 & -1 & 1\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 \end{pmatrix}$$ However, when I input this into Macaulay2, and construct a normal toric variety using this, it returns false when asked if it is smooth. However, (cf. CLS Theorem 2.4.3), we know that a projective variety corresponding to a smooth polytope must be smooth. So, how exactly did I go wrong here?

In addition, by definition, one would construct a Calabi-yau hypersurface from a reflexive polytope by taking the section of the sheaf of the anticanonical divisor, which is guaranteed to be cartier and ample, when working with Gorenstein fano toric varieties. How would one construct this explicitly in Macaulay2, starting with a polyhedra?

Finally, it is straightforward to compute the hodge number computationally as per eqn (2.1) in http://arxiv.org/pdf/1411.1418v1.pdf. How would one compute this using the built-in functions of calculating the hodge numbers?

Any explicit examples would be helpful.

Thanks!

• Can you show the Macaulay2 code? One mistake I always do is that I use the wrong polytope when I should use the polar polytope instead. Computing Hodge numbers in M2 with equations is usually hopeless. First you have to find equations for your toric variety, and then add a "random" hyperplane equation. This greatly messes up the complexity of the Gröbner basis, so almost any computation with these equations are bound to take a looong time. Commented Jul 28, 2015 at 13:05
• Thanks! It turns out I made the same mistake forgetting to use the polar polytope. Could you perhaps give an explicit example of computing the hodge number with the above polytope? Commented Jul 28, 2015 at 15:18
• You mean computing the Hodge numbers within M2? I would rather do that in Sage (which uses the formula you refer to). Commented Jul 28, 2015 at 22:46
• Would I use the function for computing it given here? doc.sagemath.org/html/en/reference/geometry/sage/geometry/… For some reason, it outputs (20,) for the hodge number. Shouldn't it output a 2-tuple corresponding to (h11,h12)? Also, I couldn't find the corresponding polytope in the database hep.itp.tuwien.ac.at/~kreuzer/CY after inputting 11 points, 10 vertex points, and 24 facets. Am I doing something wrong with the code in Sage? Commented Jul 29, 2015 at 1:30

Posting this as an answer because it is too long for comments.

Below is the SAGE sessions I used to compute.

M = matrix([[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,-1,0,0],[-1,1,0,0],[-1,0,0,0],[0,-1,0,0],[1,-1,-1,0],[-1,1,0,-1]])
P = Polyhedron(M); P
L = LatticePolytope(M.rows()); L
print L.poly_x("")
print P.f_vector()
​
A 4-dimensional polyhedron in ZZ^4 defined as the convex hull of 10 vertices
4-d reflexive polytope in 4-d lattice M
M:11 10 N:59 24 H:50,6 [88]

(1, 10, 34, 48, 24, 1)


I did this on the SAGE cloud (cloud.sagemath.com). It seems that in your comment you use the SAGE function "nef_x" which is used for complete intersection Calabi-Yaus. It computes the Hodge numbers of two divisor cuts, which in your case gives something 2-dimensional.