We introduce an elementary method to study the border rank of polynomials and tensors, analogous apolarity lemma. This can be used describe all cases uniformly, including those very special ones that resisted a systematic approach. also define version variety sums powers analyze its usefulness in studying tensors with large symmetries. In particular, it applied provide lower bounds for some int...

Backelin proved that the multigraded Poincaré series for resolving a residue field over a polynomial ring modulo a monomial ideal is a rational function. The numerator is simple, but until the recent work of Berglund there was no combinatorial formula for the denominator. Berglund’s formula gives the denominator in terms of ranks of reduced homology groups of lower intervals in a certain lattic...

We study the equations defining a projective embedding of a toric variety X using multigraded Castelnuovo-Mumford regularity. Consider globally generated line bundles B1, . . . , Bl and an ample line bundle L := B ⊗m1 1 ⊗ B2 2 ⊗ · · · ⊗ Bl l on X . This article gives sufficient conditions on mi ∈ N to guarantee that the homogeneous ideal I of X in P := P (

Generalizing cones over projective toric varieties, we present arbitrary toric varieties as quotients of quasiaffine toric varieties. Such quotient presentations correspond to groups of Weil divisors generating the topology. Groups comprising Cartier divisors define free quotients, whereas Q-Cartier divisors define geometric quotients. Each quotient presentation yields homogeneous coordinates. ...

We study the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise naturally as homogeneous coordinate rings of blow-ups of projective varieties. LetX be a projective variety over a fieldK, with homogeneous coordinate ringA. Let...

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study projective degrees rational maps . use a general framework determine map via computation multiplicity saturated special fiber ring. As specific applications, we provide explicit formulas all determined perfect ideals height two or Gorenstein three.