Finite sets of $d$-planes in affine space

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 ${\mathbb{A}}^1$.

31 pages, 8 figures

References in corpus (1)

• Finite subsets of projective space, and their ideals