Finite sets of $d$-planes in affine space
| dc.creator | Lederer, Mathias | |
| dc.date | 2008-03-21 | |
| dc.date | 2008-10-13 | |
| dc.date.accessioned | 2026-07-07T10:08:57Z | |
| dc.date.available | 2026-07-07T10:08:57Z | |
| dc.description | 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 $γ+\oplus_{j\in J}\mathbb{N}e_{j}$, where $#J=d$ and $γ_{j}=0$ 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 $A\cap\{X_{1}=λ\}$, where $λ$ runs through $\mathbb{A}^1$. | |
| dc.description | 31 pages, 8 figures | |
| dc.identifier | https://arxiv.org/abs/0803.3141 | |
| dc.identifier | http://arxiv.org/abs/0803.3141 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/171174 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 13A15, 13C05, 14N15 | |
| dc.title | Finite sets of $d$-planes in affine space | |
| dc.type | text |