Finite sets of $d$-planes in affine space

TL;DR

This paper explores the combinatorial structure of the set of exponents of standard monomials for subvarieties of affine space composed of linear or affine subspaces, linking it directly to the geometry of the variety.

Contribution

It generalizes the classical finiteness algorithm from points to higher-dimensional affine subspaces, establishing a correspondence between $d$-planes in the exponent set and components of the variety.

Findings

Number of $d$-planes in $D(A)$ equals the number of components of $A

Determines the count of all $d$-planes parallel to a given subspace

Describes $D(A)$ via standard sets of intersections with hyperplanes

Abstract

Let $A$ be a subvariety of affine space $\mathbb{A}^n$ whose irreducible components are $d$-dimensional linear or affine subspaces of $\mathbb{A}^n$. Denote by $D(A)\subset\mathbb{N}^n$ the set of exponents of standard monomials of $A$. We show that the combinatorial object $D(A)$ reflects the geometry of $A$ in a very direct way. More precisely, we define a $d$-plane in $\mathbb{N}^n$ as being a set $\gamma+\oplus_{j\in J}\mathbb{N}e_{j}$, where $#J=d$ and $\gamma_{j}=0$ for all $j\in J$. We call the $d$-plane thus defined to be parallel to $\oplus_{j\in J}\mathbb{N}e_{j}$. We show that the number of $d$-planes in $D(A)$ equals the number of components of $A$. This generalises a classical result, the finiteness algorithm, which holds in the case $d=0$. In addition to that, we determine the number of all $d$-planes in $D(A)$ parallel to $\oplus_{j\in J}\mathbb{N}e_{j}$, for all $J$. Furthermore, we describe $D(A)$ in terms of the standard sets of the intersections $A\cap\{X_{1}=\lambda\}$, where $\lambda$ runs through $\mathbb{A}^1$.