## A new ansatz for extremal Laurent polynomials

This post describes a new method for generating Laurent polynomials in 3 variables. Many of these Laurent polynomials are extremal or of low ramification, and they include the extremal Laurent polynomials mirror to 15 of the 17 minimal Fano 3-folds. We call this method the *Minkowksi ansatz*.

Let be a 3-dimensional reflexive polytope. We will construct a Laurent polynomial with Newton polytope equal to , or in other words we will explain how to assign a coefficient to each integer point in . This goes as follows.

**Lattice Minkowski sums**

We say that a polygon is the *lattice Minkowski sum* of polygons and if and only if both:

- , so that is the Minkowski sum of and as usual
- the integer lattice in is the sum of the integer lattices in and .

Note that any of the the polygons here are allowed to be degenerate.

Examples: here are two lattice Minkowksi decompositions of a hexagon:

Note that the same lattice polygon can have more than one lattice Minkowski decomposition. Note also that the first decomposition here is not a complete decomposition into lattice-Minkowksi-irreducible pieces, because the square can be further decomposed as the sum of a vertical and a horizontal line.

The example above is not a lattice Minkowski decomposition, because the lattice in is not the sum of the lattices in and . In fact is lattice Minkowski irreducible.

**Decompose the facets into irreducible pieces**

There are 4319 3-dimensional reflexive polytopes. These polytopes contain a total of 344 distinct facets, where we regard two facets as the same if and only if they differ by a lattice-preserving automorphism. Of these facets, 79 are lattice Minkowski irreducible. These 79 facets are also the non-degenerate polygons which occur when the 344 total facets are decomposed into lattice Minkowksi irreducible pieces. Of those 79 facets, exactly 8 contain no interior lattice points. Those 8 triangles, which we call *admissible triangles* are all of type :

In other words, the cones over these triangles give affine toric varieties that are transverse singularities, for .

**The ansatz**

Given a 3-dimensional reflexive polytope , we construct a possibly-empty list of Laurent polynomials as follows. For each facet of , decompose into lattice-Minkowksi-irreducible pieces in all possible ways. Discard any such decomposition of which contains a non-degenerate polygon that is not an admissible triangle. Any remaining decomposition of will consist of line segments and admissible triangles. To this decomposition we associate a Laurent polynomial which is the product of certain basic Laurent polynomials corresponding to line segments and to admissible triangles. The basic Laurent polynomials for admissible triangles are:

and so on for the remaining admissible triangles. The basic Laurent polynomials for line segments are:

and so on for other line segments.

So now, for each facet of , we have a list of Laurent polynomials; this list will be empty if cannot be written as a lattice Minkowksi sum of line segments and admissible triangles. In other words for each facet we have list of ways of assigning coefficients to each integer point in . We seek a list of Laurent polynomials with Newton polytope equal to , or in other words a list of ways of assigning coefficients to each integer point in . This is produced by assigning the coefficient zero to the origin (which is the only interior point of ) and then assigning coefficients to the integer points on facets of as specified in the facet lists (but amalgamated in all possible ways, so if there are elements in the list for facet then the number of elements in the list for is ).

**Points to Note**

- This ansatz almost generalizes the earlier recipes given by Pryjzalkowski and Galkin, but differs a little because of the difference between Minkowski decomposition and lattice Minkowksi decomposition.
- Altman has studied the deformation theory of affine toric varieties and discovered a close connection with Minkowski decompositions. Since we expect to find the local system associated to an extremal Laurent polynomial as a piece of the quantum cohomology local system associated to a smoothing of the Newton polytope of , this is encouraging. But note that Minkowski decomposition and lattice Minkowksi decomposition are not the same.
- We suspect that if is a 3-dimensional reflexive polytope containing a facet with no admissible lattice Minkowski decompositions then the toric variety corresponding to
*does not smooth*. More on this later. - This ansatz also fits well with Kouchnirenko’s criterion for a Laurent polynomial to be degenerate.

(I learned this last point from Hiroshi.)

You mean the singularity corresponding to the cone over this facet does not admit a smoothing?

The ansatz looks reasonable, since it obviously respects the simplest type of the symplectomorphisms

where

and

the product corresponds to some Minkowski decomposition of the facet supporting i.e. and .

I expect there is an ansatz like that for all

smoothablepolytopes (maybe not Gorenstein).Note that my original post contained a mistake which I have now corrected: the only change is the description of how to go from a collection of lists of Laurent polynomials, one for each facet, to a list of Laurent polynomials for the polytope .

Sergei: does your comment 2 still apply to this corrected version?

@Sergei: with regard to comment 1, what Alessio expects is that if a 3d reflexive polytope contains a face without admissible decompositions then the toric variety corresponding to is not smoothable. This is precisely because, as you suggest, he expects that the toric co-ordinate patch on corresponding to the cone over will not smooth. But even if each face of is such that the cone over it gives a smoothable affine toric variety, I expect that there may still be some global obstruction to smoothability because the local smoothings (i.e. the smoothings of the toric co-ordinate patches) may not glue together to give a smoothing of the whole toric variety.

I cannot see what was the typo, but my comment2 is a positive feedback to the ansatz you wrote now — if there are two parallel facets, you can transmit some of their divisors between them.

On local-to-global obstructions to smoothability: in dimension 2 they vanish, for ODP’s in dimension 3 they also vanish.

There are some questions:

Q1. a) What are the smoothable normal 3-dimensional toric singularities?

There are some isolatedand some germs of non-isolated

b) What are the possible incidence relations between them?

Q2. Are local-to-global obstructions to smoothability vanish for smoothable toric singularities in dimension 3?

Q3. Show (or disprove) that

if

the affine singularities corresponding to the cones over and are smoothable

then

the affine singularity corresponding to the cone over the Minkowski sum is smoothable?

According to Prokhorov Q1 is open.

Answer for Q3 could be in Altmann-Straten.

I corrected a typo (343 distinct facets->344 distinct facets, twice) pointed out by Al. Sorry!