Smooth Fano Polytopes and Hypersurfaces

Question

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!

Answer 1

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.