StronglyStableIdeals

Macaulay2 package StronglyStableIdeals -- Find strongly stable ideals with a given Hilbert polynomial. Strongly stable ideals are a key tool in commutative algebra and algebraic geometry. These ideals have nice combinatorial properties that make them well suited for both theoretical and computational applications. In the case of polynomial rings with coefficients in a field of characteristic zero, the notion of strongly stable ideals coincides with the notion of Borel-fixed ideals. Borel-fixed ideals are fixed by the action of the Borel subgroup of triangular matrices and a famous result by Galligo says that generic initial ideals are of this type. In the context of Hilbert schemes, Galligo’s theorem means that each component and each intersection of components contains at least a point corresponding to a scheme defined by a Borel-fixed ideal. Hence, these ideals are distributed throughout the Hilbert schemes and they can be used to understand its local structure. The main feature of the this package is a method to compute the set of all saturated strongly stable ideals in a given polynomial ring with a given Hilbert polynomial