We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant coefficient case. Generically the multiplicity of the ideal is shown to correspond directly to a natural definition of multiplicity of cones of tropical varieties. Moreover, we can recover information on ...

We study (slope-)stability properties of syzygy bundles on a projective space P given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical f...

Quasi-stable ideals appear as leading ideals in the theory of Pommaret bases. We show that quasi-stable leading ideals share many of the properties of the generic initial ideal. In contrast to genericity, quasistability is a characteristic independent property that can be effectively verified. We also relate Pommaret bases to some invariants associated with local cohomology, exhibit the existen...

Let A = K[x1, . . . , xn] denote the polynomial ring in n variables over a field K of characteristic 0 with each deg xi = 1. Given arbitrary integers i and j with 2 ≤ i ≤ n and 3 ≤ j ≤ n, we will construct a monomial ideal I ⊂ A such that (i) βk(I) < βk(Gin(I)) for all k < i, (ii) βi(I) = βi(Gin(I)), (iii) βl(Gin(I)) < βl(Lex(I)) for all l < j and (iv) βj(Gin(I)) = βj(Lex(I)), where Gin(I) is t...