The Hilbert polynomial can be defined axiomatically as a group homomorphism on the Grothendieck group K(X) of a projective variety X, satisfying certain properties. The Chern polynomial can be similarly defined. We therefore define these rather abstract notions to try and find a nice description of this relationship. Introduction Let X = P be a complex projective variety over an algebraically c...

The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert–Serre). We derive an upper bound for its grade, i. e. the index from which on its coefficients are constant. As an application, we give a purely algebraic proof of an old combinatorial result (due to Ehrhart, McMullen and Stanley). 1. HILBERT QUASIPOLYNOMIALS Let K be a field, and R a positive...

Contents 1 Conventions and basic notation 2 2 Hilbert variety: the general framework 3 3 The Hilbert curve 8 4 Cubic Hilbert curves 12 5 Image of the Hilbert curve in P 3 17 6 Fibrations and singular points of the Hilbert curve at infinity 19 7 Serre-invariant curves 23 Abstract Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V X associated to the Hilbe...

The Grothendieck-Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two m-primary ideals I and J in a local ring (A, m) in terms of local cohomology modules of Rees algebras ...

We prove that the F-signature of an affine semigroup ring of positive characteristic is always a rational number, and describe a method for computing this number. We use this method to determine the F-signature of Segre products of polynomial rings, and of Veronese subrings of polynomial rings. Our technique involves expressing the F-signature of an affine semigroup ring as the difference of th...

It is shown that the Hilbert series of the face ring of a clique complex (equivalently, flag complex) of a graph G is, up to a factor, just a specialization of S G (x, y), the subgraph polynomial of the complement of G. We also find a simple relationship between the size of a minimum vertex cover of a graph G and its subgraph polynomial. This yields a formula for the h-vector of the flag comple...