HILBERT · Gröbner strata in multigraded Hilbert schemes

The aim of the present research project is to establish new connections between algebraic geometry, commutative algebra and combinatorics. The geometric objects of study are Hilbert schemes. These schemes are highly relevant in algebraic geometry, as they form the basis for the construction of numerous moduli spaces. The algebraic objects of study are Gröbner bases. They are the core of great parts of constructive methods in commutative algebra. The combinatorial objects of study are standard sets. These are central to the theory of Gröbner bases, as there is a canonical bijection between monomial ideals and standard sets. I use a newly defined addition of standard sets, which establishes a link between geometry, algebra, and combinatorics. I will construct a new moduli space which parametrises all reduced Gröbner bases in a polynomial ring having a prescribed standard set. I will embed this moduli space as a locally closed subscheme into various multigraded Hilbert schemes after Haiman and Sturmfels, and decompose a given multigraded Hilbert scheme as a coproduct of moduli spaces of reduced Gröbner bases, where the union is indexed by a set of standard sets. Moreover, I will pursue the question whether the above described decomposition into locally closed strata is a stratification of the multigraded Hilbert scheme or not. In the case where the given standard set is finite, I have already constructed the moduli space of all reduced Gröbner bases with the given standard set. I have aldready embedded this moduli space canonically into the Hilbert scheme of points. Now I will study these spaces in more detail, using the lexicographic order on the polynomial ring. I shall prove a conjecture by Sturmfels which establishes a close connection between the geometry of moduli spaces of reduced Gröbner bases on the one hand and the combinatorics of standard sets on the other.