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.

中文翻译：

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奇异性和激进的初始理想

作为光滑射影变种的Gröbner退化，可以得到哪种简化的单项式方案？我们的猜想答案是：仅与无环Cohen-Macaulay简单复形相关的Stanley-Reisner方案。特别地，这意味着只有零属的曲线具有这种退化。我们证明了这个猜想是针对degrevlex阶，针对完全实数域的椭圆曲线，针对交叉多边形的边界以及针对无叶图。我们用矫正定律讨论代数有理和F有理奇异性的后果。