We investigate the Koszul property for quotients of affine semigroup rings by semigroup ideals. Using a combinatorial and topological interpretation for the Koszul property in this context, we recover known results asserting that certain of these rings are Koszul. In the process, we prove a stronger fact, suggesting a more general definition of Koszul rings. This more general definition of Koszulness turns out to be satisfied by all Cohen-Macaulay rings of minimal multiplicity.

Abstract. The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the ℓ-adic cohomology of the arrangement (as a variety). We describe the eigenvalues of the Frobenius map acting on this cohomology, which corresponds to a finer decomposition of the zeta function. The ℓ-adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data. Finally, it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases. 1.

Abstract. The well-known “splitting necklace theorem ” of Alon [1] says that each necklace with k · ai beads of color i = 1,..., n can be fairly divided between k “thieves ” by at most n(k − 1) cuts. Alon deduced this result from the fact that such a division is possible also in the case of a continuous necklace [0, 1] where beads of given color are interpreted as measurable sets Ai ⊂ [0, 1] (or more generally as continuous measures µi). We demonstrate that Alon’s result is a special case of a multidimensional, consensus division theorem of n continuous probability measures µ1,..., µn on a d-cube [0,1] d. The dissection is performed by m1 +... + md = n(k − 1) hyperplanes parallel to the sides of [0,1] d dividing the cube into m1 ·... · md elementary parallelepipeds where the integers mi are prescribed in advance. 1.