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., $x_0^n=x_1 \cdots x_n$.

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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 projectively normal toric variety is Koszul.