On quadratic persistence and Pythagoras numbers of totally real projective varieties

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.