BOUNDS ON HILBERT FUNCTIONS OF SUBALGEBRAS IN DEGREE TWO

Given a subspace U⊂ℂ[x1,… ⁡,xn]d we consider the closure of the image of the rational map ℙn−1 −−→ℙdim U−1 given by U. Its coordinate ring is isomorphic to ⊕ ⁡i≥0Ui, where Ui 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.