Singularities and radical initial ideals. Issue 4 (28th June 2020) - Constantinescu, Alexandru; De Negri, Emanuela; Varbaro, Matteo
Abstract: What kind of reduced monomial schemes can be obtained as a Gröbner degeneration of a smooth projective variety? Our conjectured answer is: only Stanley–Reisner schemes associated to acyclic Cohen–Macaulay simplicial complexes. This would imply, in particular, that only curves of genus zero have such a degeneration. We prove this conjecture for degrevlex orders, for elliptic curves over totally real number fields, for boundaries of cross‐polytopes, and for leafless graphs. We discuss consequences for rational and F‐rational singularities of algebras with straightening laws.