Locally unmixed degrees

(with Amit Khetan)

        

n - dimensional polytopes  P1 , . . . , Pk  in  Rn  are said to share a complete flag of faces if there are facets  Fi  of  Pi  with the same outer normal and (after translation to the same hyperplane) the (n - 1) - dimensional polytopes  F1 , . . . , Fk  share a complete flag of faces. We assume that 0-dimensional polytopes always share a complete flag of faces.

For example, the four 3-dimensional polytopes above share a complete flag of faces: each of them has a face with outer normal (0, 0, 1) and this face has an edge with outer normal (-1, 0) (in the plane z = 0).

Now let  X  be a complete  n-dimensional toric variety and  d0 , . . . , dn  be  n + 1 semi-ample degrees on  X  with polytopes  P0 , . . . , Pn . We are interested in computing the toric residue map of  n + 1 sections  f0 , . . . , fn  of the corresponding degrees. It turns out that this problem can be reduced to finding a collection of partitions of the vertices of the polytopes satisfying a simple compatibility property.

Compatible partitions can be found, for example, when the polytopes share a complete flag of faces. In this case we call the degrees locally unmixed. This is a broad generalization of the unmixed case when the polytopes are the same (or more generally when they share the normal fan).

It is not true that any collection of polytopes has a (non-trivial) compatible partition of its vertices. However we hope that for generically positioned polytopes (one needs to define what generic means) compatible partitions exist.

If curious you can find the details in our preprint.

Compatible partitions:

Let  P0 , . . . , Pn  be polytopes in  Rn . Assume the vertices of each polytope  Pi  are partitioned into  n + 1 subsets: Vi 0 , . . . , Vi n .

The partitions are compatible if for any permutation  S  of  {0 , . . . , n}  each sum

v0 + . . . + vn,      where   vi   is a vertex from   Vi S(i)

lies in the interior of the Minkowski sum  P0 + . . . + Pn .