The reduction number and degree bound of projective subschemes

TL;DR

This paper establishes degree bounds for projective subschemes based on the reduction number, characterizes maximal cases as arithmetically Cohen-Macaulay with linear resolution, and classifies varieties with almost maximal degree.

Contribution

It introduces new degree bounds linked to the reduction number and characterizes maximal and almost maximal degree cases with explicit Betti tables.

Findings

Degree upper bounds are proven in terms of the reduction number.

Maximal cases are arithmetically Cohen-Macaulay with linear resolution.

Only two types of reduced, irreducible varieties have almost maximal degree.

Abstract

In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with linear resolution. Furthermore, it can be shown that there are only two types of reduced, irreducible projective varieties with almost maximal degree. We also give explicit Betti tables for almost maximal cases. Interesting examples are provided to understand our main results.