Bounds on Hilbert functions with application to convexity

Given a subspace $U\subset\mathbb{C}[x_1,\dots,x_n]_d$ we consider the closure of the image of the rational map $\mathbb{P}^{n-1}\dashrightarrow\mathbb{P}^{\dim U-1}$ given by $U$. Its coordinate ring is isomorphic to $\bigoplus_{i\ge 0} U^i$ where $U^i$ is the degree $i$ component. We consider the Hilbert function of this algebra in the case where $U$ contains a regular sequence, equivalently the map is a morphism, and find lower bounds for the dimension of the degree 2 component. We apply our bounds to study the boundary structure of certain convex sets, called Gram spectrahedra, which are linked to sum of squares representations of non-negative polynomials.