Initial simplicial complexes of prime ideals

We provide a two-parameter family of examples of irreducible projective algebraic varieties whose initial complexes (in the sense of [M. Kalkbrener, B. Sturmfels, Adv. Math. 116 (1995) 365–376]) have the maximum number of simplices given the dimensions of the variety and of its ambient projective space. This shows that irreducibility fails to be preserved in the worst possible fashion by the operation of passing to the Stanley–Reisner variety of the initial complex.