# Componentwise linearity of projective varieties with almost maximal degree

## TL;DR

This paper investigates projective varieties with degrees just below the maximum, showing that most such varieties have componentwise linear ideals, which helps classify their Betti tables and compute their resolutions.

## Contribution

It demonstrates that most varieties with almost maximal degree have componentwise linear ideals, enabling explicit classification of their Betti tables.

## Key findings

- Most varieties with almost maximal degree are componentwise linear.
- Betti tables of these varieties can be explicitly computed.
- Componentwise linearity aids in classifying Betti tables.

## Abstract

The degree of a projective subscheme has an upper bound in term of the codimension and the reduction number. If a projective variety has an almost maximal degree, that is, the degree equals to the upper bound minus one, then its Betti table has been described explicitly. We build on this work by showing that for most of such varieties, the defining ideals are componentwise linear and in particular the componentwise linearity is suitable for classifying the Betti tables of such varieties. As an application, we compute the Betti table of all varieties with almost maximal degree and componentwise linear resolution.