More Algebraic Geometry posted

**topspin1617** · February 11th 2011, 11:11 PM

Again, I know this one looks like it should be easy (easier, in fact, even than what I posted before). My brain is just completely fried on this subject right now for some reason...

A hyperplane in $\mathbb{P}^N$ is a hypersurface [which means its dimension (as a topological space in the Zariski topology on $\mathbb{P}^N)$ is $\dim\mathbb{P}^N-1=N-1$ ], which is defined by a single linear polynomial. That is, our hyperplane is $H=\mathcal{Z}(f)\subseteq\mathbb{P}^N$ , where $f$ is a linear, homogeneous (since we are in projective space) polynomial in $S=k[x_0,x_1,\ldots,x_N]$ ( $k$ an algebraically closed field) . The problem says to determine that $\mathbb{P}^N\setminus H$ is an affine variety.

**topspin1617** · February 12th 2011, 10:50 AM

Okay... I'm really not trying to beat this thing to death, but I have a preliminary exam coming up soon, and I'm very nervous because I still feeling like I'm struggling with what should be the most basic of questions.

I DO know that, if the hyperplane is defined simply by one of the coordinates (for sake of example, say $H=\mathcal{Z}(x_0)$ ), then the statement is true. On the complement of $H$ , we know that the $x_0$ coordinate is nonzero. Since we are in projective space, we may assume that $x_0=1$ (because in $\mathbb{P}^n,(x_0,\ldots,x_n)\equiv (\lambda x_0,\ldots,\lambda x_n)\, \forall \, \lambda\in k\setminus\{0\}$ ; if we are restricting to points where $x_0\neq 0$ , we may as well "normalize" our points by dividing through by $x_0$ ). In this case, after this normalization, we may define a map

$\varphi:\mathbb{P}^n\setminus H\rightarrow\mathbb{A}^n$
$(1,x_1,\ldots,x_n)\mapsto(x_1\ldots,x_n)\in\mathbb {A}^n$

This turns out to be an isomorphism $\mathbb{P}^n\setminus H\cong\mathbb{A}^n$ .

Is there some sort of way I can generalize this to the complement of an arbitrary hyperplane (that is, to a set defined by the equation $a_0x_0+\cdots+a_nx_n\neq 0$ ,each $a_i\in k$ )? Maybe some sort of (projective..) linear change of coordinates (which I don't know how to do...)? Would the mapping, for example, $x_0\mapsto f,x_1\mapsto x_1,\ldots,x_n\mapsto x_n$ be a legal isomorphism on the points of $\mathbb{P}^n$ , in which case I could apply the idea above? Or is there something completely different I should be looking at?

I'm sorry for posting this stuff over and over when it seems no one really is able to help anyway... but I'm.. well, don't wanna say desperate, but... yea. I really feel like I am understanding the concepts, but I'm not able to apply them at all to solve anything.

EDIT: Am I asking this in the appropriate forum, or would another section be more appropriate? I just figure it would go here since it uses a good deal of abstract algebra...

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