Betti Numbers and Degree Bounds for Some Linked Zero-Schemes

TL;DR

This paper investigates the relationship between Betti numbers and degree bounds of linked zero-schemes, extending known bounds through linkage techniques, especially for Cohen-Macaulay cases and residual schemes.

Contribution

It demonstrates the conjectured degree bound holds for certain linked zero-schemes, particularly when the schemes are residual to low-degree or specially positioned zero-schemes.

Findings

Degree bounds hold for residual schemes in specific cases

Linkage reduces the problem to zero-dimensional subschemes

Results extend previous bounds to new classes of schemes

Abstract

In their paper on multiplicity bounds (1998), Herzog and Srinivasan study the relationship between the graded Betti numbers of a homogeneous ideal I in a polynomial ring R and the degree of I. For certain classes of ideals, they prove a bound on the degree in terms of the largest and smallest Betti numbers, generalizing results of Huneke and Miller (1985). The bound is conjectured to hold in general; we study this using linkage. If R/I is Cohen-Macaulay, we may reduce to the case where I defines a zero-dimensional subscheme Y. If Y is residual to a zero-scheme Z of a certain type (low degree or points in special position), then we show that the conjecture is true for I_Y.