Given multigraded free resolutions of two monomial ideals we construct a multigraded free resolution of the sum of the two ideals. Introduction Let K be a field, S = K[x1, . . . , xn] the polynomial ring in n variables over K, and let I and J be two monomial ideals in S. Suppose that F is a multigraded free S-resolution of S/I and G a multigraded free S-resolution of S/J . In this note we const...

This paper introduces a class of smooth projective varieties that generalise and share many properties with partial flag varieties of type A. The quiver flag variety Mθ(Q, r) of a finite acyclic quiver Q (with a unique source) and a dimension vector r is a fine moduli space of stable representations of Q. Quiver flag varieties are Mori Dream Spaces, they are obtained via a tower of Grassmann bu...

We develop a theory of multigraded (i.e., N-graded) combinatorial Hopf algebras modeled on the theory of graded combinatorial Hopf algebras developed by Aguiar et al. (Compos. Math. 142:1–30, 2006). In particular we introduce the notion of canonical k-odd and k-even subalgebras associated with any multigraded combinatorial Hopf algebra, extending simultaneously the work of Aguiar et al. and Ehr...

In this note we develop some of the properties of separators of points in a multiprojective space. In particular, we prove multigraded analogs of results of Geramita, Maroscia, and Roberts relating the Hilbert function of X and X \ {P} via the degree of a separator, and Abrescia, Bazzotti, and Marino relating the degree of a separator to shifts in the minimal multigraded free resolution of the ...

We study unitary multigraded non-associative algebras R generated by an ordered set X over a field K of characteristic 0 such that the mappings ∂k : xl → δkl, xk, xl ∈ X, can be extended to derivations of R. The class of these algebras is quite large and includes free associative and Jordan algebras, absolutely free (non-associative) algebras, relatively free algebras in varieties of algebras, ...

We introduce and study the toric Hilbert scheme that parametrizes all ideals with the same multigraded Hilbert function as a given toric ideal.

Given a multigraded algebra A, it is a natural question whether or not for two homogeneous components Au and Av, the product AnuAnv is the whole component Anu+nv for n big enough. We give combinatorial and geometric answers to this question. 1. Statement and discussion of the results In this note, we consider the multiplication map of a multigraded algebra and ask for its surjectivity propertie...