We show that the Hilbert scheme, that parametrizes all ideals with the same Hilbert function over a Clements-Lindtröm ring W, is connected. More precisely, we prove that every graded ideal is connected by a sequence of deformations to the lex-plus-powers ideal with the same Hilbert function. Our result is an analogue of Hartshorne’s theorem that Grothendieck’s Hilbert scheme is connected; howev...

The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization exact categories triangulated categories. Our first aim is to provide an extension Auslander's formula: for some $\mathcal{C}$, there exists localization sequence $\operatorname{def}\mathcal{C} \to \mod\mathcal{C} \operatorname{lex}\mathcal{C}$, where $\operatorname{lex}\mathcal{C}$ den...

Logical models of legal reasoning often assume that norms give a complete description of their applicability conditions, an assumption which is too strong, due to the complexities and dynamics of the world. Norms cannot take into account all the possible conditions where they should or should not be applied, first of all because the legislator cannot consider all the possible contexts which are...

We study the Betti numbers of graded ideals containing the squares of the variables, in a polynomial ring. We prove the lex-plus-powers conjecture for such ideals.

We prove the almost complete intersection case of the Lex-Plus-Powers Conjecture on graded Betti numbers. We show that the resolution of a lex-plus-powers almost complete intersection provides an upper bound for the graded Betti numbers of any other ideal with regular sequence in the same degrees and the same Hilbert function. A key ingredient is finding an explicit comparison map between two K...