On quadratic persistence and Pythagoras numbers of totally real projective varieties

TL;DR

This paper explores the relationship between quadratic persistence and Pythagoras numbers in totally real projective varieties, extending previous characterizations to broader classes and classifying certain varieties with specific properties.

Contribution

It extends the classification of varieties with next-to-maximal quadratic persistence beyond arithmetically Cohen-Macaulay cases, covering more general totally real varieties.

Findings

Classified totally real non-aCM varieties with next-to-maximal quadratic persistence for specific codimension and degree ranges.

Analyzed quadratic persistence and Pythagoras number for curves of maximal regularity.

Provided new insights into the structure of linearly normal smooth curves of genus 3.

Abstract

In this paper, we study the relationship between quadratic persistence and the Pythagoras number of totally real projective varieties. Building upon the foundational work of Blekherman et al. in arXiv:1902.02754, we extend their characterizations of arithmetically Cohen-Macaulay varieties with next-to-maximal quadratic persistence to arbitrary case. Our main result classifies totally real non-aCM varieties of codimension $c$ and degree $d$ that exhibit next-to-maximal quadratic persistence in the cases where $c=3$ and $d \geq 6$ or $c \geq 4$ and $d \geq 2c+3$. We further investigate the quadratic persistence and Pythagoras number in the context of curves of maximal regularity and linearly normal smooth curves of genus 3.