For totally real varieties of codimension 3 (degree at least 6) or higher codimension with large degree, next-to-maximal quadratic persistence is equivalent to lying as a codimension-1 subvariety in a variety of minimal degree.
Pythagoras Numbers for Ternary Forms
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abstract
We study the Pythagoras numbers $py(3,2d)$ of real ternary forms, defined for each degree $2d$ as the minimal number $r$ such that every degree $2d$ ternary form which is a sum of squares can be written as the sum of at most $r$ squares of degree $d$ forms. Scheiderer showed that $d+1\leq py(3,2d)\leq d+2$. We show that $py(3,2d) = d+1$ for $2d = 8,10,12$. The main technical tool is Diesel's characterization of height 3 Gorenstein algebras.
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On quadratic persistence and Pythagoras numbers of totally real projective varieties
For totally real varieties of codimension 3 (degree at least 6) or higher codimension with large degree, next-to-maximal quadratic persistence is equivalent to lying as a codimension-1 subvariety in a variety of minimal degree.