We show that for a smooth hypersurface X ⊂ P of degree at least 2, there exist arithmetically Cohen-Macaulay (ACM) codimension two subvarieties Y ⊂ X which are not an intersection X ∩ S for a codimension two subvariety S ⊂ P. We also show there exist Y ⊂ X as above for which the normal bundle sequence for the inclusion Y ⊂ X ⊂ P does not split. Dedicated to Spencer Bloch

The connection between Feynman integrals and GKZ $A$-hypergeometric systems has been a topic of recent interest with advances in mathematical techniques computational tools opening new possibilities; this paper we continue to explore connection. To each such hypergeometric system there is an associated toric ideal, prove that the latter Cohen-Macaulay property for two large families integrals. ...