The following question was posed by Rikard Bögvad in the paper On the homogeneous ideal of a projective nonsingular toric variety:
Is the toric ideal of a smooth projectively normal toric variety generated by quadrics?
This is interesting, since toric ideals have an explicit description. In particular, it is not known if the coordinate ring of a smooth projectively normal toric variety is Koszul. Smoothness is of course essential here, since there are many toric hypersurfaces of degree $\ge 3$, e.g., $x_0^n=x_1 \cdots x_n$.