The present paper shall provide a framework for working with Gröbner bases over arbitrary rings k with a prescribed finite standard set ∆. We show that the functor associating to a k-algebra B the set of all reduced Gröbner bases with standard set ∆ is representable and that the representing scheme is a locally closed stratum in the Hilbert scheme of points. We cover the Hilbert scheme of points by open affine subschemes which represent the functor associating to a k-algebra B the set of all border bases with standard set ∆ and give reasonably small sets of equations defining these schemes. We show that the schemes parametrizing Gröbner bases are connected; give a connectedness criterion for the schemes parametrizing border bases; and prove that the decomposition of the Hilbert scheme of points into the locally closed strata parametrizing Gröbner bases is not a stratification.

Given a finite set of closed rational points of affine space over a field, we give a Gröbner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger-Möller algorithm, but in contrast to that, we determine the set of leading terms of the ideal without solving any linear equation but by induction over the dimension of affine space. The elements of the Gröbner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.

The ring of symmetric functions (or a biRees algebra thereof) can be implemented in the homology of a,b Gr a (C a+b ), the multiplicative structure being defined from the "direct sum" map. There is a natural circle action (simultaneously on all Grassmannians) under which each direct sum map is equivariant. Upon replacing usual homology by equivariant K-homology, we obtain a 2-parameter deformation of the ring of symmetric functions.This ring has a module basis given by Schubert classes [X λ ]. Geometric considerations show that multiplication of Schubert classes has positive coefficients, in an appropriate sense. In this paper we give manifestly positive formulae for these coefficients: they count numbers of "DS pipe dreams" with prescribed edge labelings. CONTENTS 1. Introduction, and statement of results 1.1. The homology ring 1.2. A two-parameter deformation 1.3. Pipe dreams for R H S 1.4. Pipe dreams for R K S 1.5. Outline of the paper 2. Interval positroid varieties and IP pipe dreams 2.1. Positroid varieties, interval positroid varieties, and duality 2.2. The K T -formula from [Kn, §4.4] 3. Degeneration in stages, of direct sum varieties 3.1. The positroid variety to start with 3.2. The bottom b + d + c rows, of which d + c are trivial 3.3. Duality at mid-sort 3.4. The final a rows 3.5. Proof of theorem 1.4 References Date: March 9, 2015.

We fix the lexicographic order ≺ on the polynomial ring S=k[x1, …, xn] over a ring k. We define HilbS/k≺Δ, the moduli space of reduced Gröbner bases with a given finite standard set Δ, and its open subscheme HilbS/k≺Δ,ét, the moduli space of families of # Δ points whose attached ideal has the standard set Δ. We determine the number of irreducible and connected components of the latter scheme; we show that it is equidimensional over Spec k; and we determine its relative dimension over Spec k. We show that analogous statements do not hold for the scheme HilbS/k≺Δ. Our results prove a version of a conjecture by Bernd Sturmfels.

From Kostant's multiplicity formula for general linear groups, one can derive a formula for the Kostka numbers. In this note we give a combinatorial proof of this formula.