Strongly stable ideals and Hilbert polynomials

Strongly stable ideals are a key tool in commutative algebra and algebraic geometry. These ideals have nice combinatorial properties that makes them well suited for both theoretical and computational applications. In the case of polynomial rings with coefficients in a field with characteristic zero, the notion of strongly stable ideals coincides with the notion of Borel-fixed ideals. Ideals of the latter type are fixed by the action of the Borel subgroup of triangular matrices of the linear group and they correspond to the possible generic initial ideals by a famous result by Galligo [Gal74]. In the context of Hilbert schemes, Galligo’s theorem states 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 represent a promising tool for studying Hilbert schemes and to obtain a complete knowledge of their structure. To this aim, in recent years several authors [LR11, CR11, BCLR13, BLR13, LR13] developed algorithmic methods based on the use of strongly stable ideals to construct flat families corresponding to open subschemes of a Hilbert scheme. Once we are able to determine such open subschemes, in order to study all the components of a Hilbert scheme, we need to determine all its points corresponding to schemes defined by Borel-fixed ideals, i.e. all the saturated strongly stable ideals in a polynomial ring with a given Hilbert polynomial. The main feature of the package StronglyStableIdeals is a method to compute this set of ideals introduced in [CLMR11] and improved in [Lel12]. Several other tools are developed and presented in the paper.