i have just started to learn some algebraic geometry and there is a statement in the notes i am following that i do not understand: "Subvector spaces of $\mathbb{A}^n$ are algebraic sets. They are of the form $Z(f_1,\ldots,f_n)$ where $f_i$ are polynomials of degree 1." (from notes by Lothar Goettsche: http://users.ictp.it/~gottsche/)

My problem with this statement is the following: For every point $a =(a_1,\ldots, a_n)$ in $A^n$ i can give a set $S$ of polynomials such that their common zero set ist $\{a\}$: $S = \{x_1-a_1, \ldots, x_n - a_n\}$. But how can i find a set of polynomials that vanishes on a whole vector space ? Of course one can take the zero polynomial, but that would not be of degree one ?

Thank you for advice


  • $\begingroup$ You have probably learned in your linear algebra course how to describe linear subspaces by equations... $\endgroup$ Mar 9 '12 at 8:41
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    $\begingroup$ " But how can i find a set of polynomials that vanishes on a whole vector space ? " For example, in $\mathbb C^3$ with coordinates $x,y,z$ , all polynomials of the form $ay+bz$ vanish on the $x$-axis, which is a sub-vector space of $\mathbb A^3(\mathbb C)=\mathbb C^3$. $\endgroup$ Mar 9 '12 at 8:43
  • $\begingroup$ I thought affine space wasn't equipped with a vector space structure? The definition I'm used to is that $\mathbb A_k^n = k^n $ as sets, and that the structure we place on $\mathbb A_k^n $ is a topological one and, in particular, there is no 'distinguished point' i.e. no origin. $\endgroup$
    – Matt
    Mar 9 '12 at 16:48

A sub vector space is an intersection of hyperplanes. Every hyperplane is the zero set of a linear form (hence a polynomial of degree 1). The common zero set of all these linear forms (which by definition is an algebraic set) is you sub vector space.

  • $\begingroup$ Cher QiL, hyperplan est féminin en anglais, donc hyperplane :-) $\endgroup$ Mar 9 '12 at 9:12
  • $\begingroup$ @GeorgesElencwajg: Oui, je vais arrêter d'écrire in English ! Merci pour le moyen mnémotechnique. $\endgroup$
    – user18119
    Mar 9 '12 at 9:19
  • $\begingroup$ I will try that. Lets say, i want to describe a two dimensional subspace of C^3. ax is zero on the y axis and on the z axis. Thus it is zero on the two dimensional subspace spanned by those two axis. Ist this the way to think about it ? $\endgroup$ Mar 9 '12 at 9:27
  • $\begingroup$ @readingframe: oui. $\endgroup$
    – user18119
    Mar 9 '12 at 9:33

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