On Quadratic Persistence and the Pythagoras Number of Projective Varieties

The Pythagoras number of a projective variety is defined as the minimum number of squares required to represent (nonnegative) quadratic forms on the real homogeneous coordinate ring of the variety as a sum of squares. It was introduced by Blekherman, Sinn, Smith, and Velasco in 2022. To investigate this semi-algebraic quantity, they introduced and analyzed various algebraic invariants, including quadratic persistence, that establish bounds on the Pythagoras number. In particular, they found intriguing equivalences between algebraic and semi-algebraic invariants for some classes of varieties, such as varieties of minimal degree or arithmetically Cohen-Macaulay varieties of almost minimal degree, known as del Pezzo varieties.

We investigate quadratic persistence and Pythagoras numbers in the next class of varieties after those listed above, including some non-arithmetically Cohen-Macaulay varieties and those where the quadratic persistence equals the codimension minus two. We mainly focus on curves C such that deg(C) = codim(C) + 3 in this talk, but we will discuss further variety classes if time permits. This talk is based on joint work with Euisung Park and Jong In Han.