0

I am reading Gathmann's notes on Algebraic Geometry https://agag-gathmann.math.rptu.de/class/alggeom-2014/alggeom-2014-c6.pdf and he defines projective n-space to be the set of 1-dimensional subspaces of $K^{n+1}$. I initially thought that it might be easier to remember if defining the projective n-space to be 1-dimensional subspaces of $K^n$. What are some good reasons to define it to be 1-dimensional subspaces of $K^{n+1}$? Is it because in this case, projective 1-space actually makes sense (not trivial)?

Also, is it that when we talk about projective n-space, n is usually a positive number (not including $0$)?

Thank you!

Coco
  • 758
  • 4
    It’s because you want the projective $n$-space to be $n$ dimensional. – K. Makabre Apr 30 '23 at 23:44
  • Yes, it is for dimension reasons. – Randall Apr 30 '23 at 23:45
  • A circle is one dimensional but we normally draw it in two dimensions. It's the same idea with projective spaces. – CyclotomicField Apr 30 '23 at 23:49
  • Maybe could be useful take a look of https://math.stackexchange.com/questions/85394/why-the-emphasis-on-projective-space-in-algebraic-geometry – sti9111 May 01 '23 at 13:59
  • Thank you! @DankaMakabre Another question, do we consider projective $0$-space? Is it that we don't consider it because it's too trivial (i.e. $n$ in projective $n$-space is always a positive integer)? – Coco May 02 '23 at 03:32

1 Answers1

0

It’s because you want the projective $n$-space to be $n$ dimensional. – Danka Makabre

Hank Scorpio
  • 2,951