Bipolynomial Hilbert functions

TL;DR

This paper introduces the concept of bipolynomial Hilbert functions for certain algebraic schemes and proves it for schemes consisting of a plane and generic lines, also conjecturing a broader applicability.

Contribution

It defines bipolynomial Hilbert functions and proves this property for schemes with a plane and generic lines, proposing a conjecture for more general configurations.

Findings

X with a plane and generic lines has bipolynomial Hilbert function

Conjecture: generic non-intersecting linear spaces also have bipolynomial Hilbert function

Provides a new perspective on Hilbert functions of algebraic schemes

Abstract

Let X be a closed subscheme and let HF(X,-) and hp(X,-) denote, respectively, the Hilbert function and the Hilbert polynomial of X. We say that X has bipolynomial Hilbert function if HF(X,d)=min{hp(P^n,d),hp(X,d)} for every non-negative integer d. We show that if X consists of a plane and generic lines, then X has bipolynomial Hilbert function. We also conjecture that generic configurations of non-intersecting linear spaces have bipolynomial Hilbert function.