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I am looking for clarification for the following problem. The field $k$ is algebraically closed and has characteristic not equal to $2.$ :

Show that the locus of all homogeneous polynomials of degree $d$ in $n+1$ variables over $k$ is a projective variety.

It seems to me that instead of considering the set $S$ of all homogeneous polynomials of degree $d$ in $n+1$ variables, we should consider $S/\sim$ where $f \sim g \in S$ if and only if there exists some nonzero scalar $a \in k$ so that $af = g.$ Here is what I have tried that has led me to this conclusion:

Proof. Let $L$ consist of all homogeneous polynomials of degree $d$ in $n+1$ variables over $k$ as well as the zero polynomial. Then $L$ is an $\binom{n+d}{d}$ dimensional vector space. There is a vector space isomorphism $L \to k^{\binom{n+d}{d}}$ given by mapping a polynomial $f$ to the $\binom{n+d}{d}$-tuple consisting of the coefficients of $f$. This isomorphism identifies one dimensional linear subspaces of $L$ with one dimensional linear subspaces of $k^{\binom{n+d}{d}}.$ That is, $S/\sim \cong \mathbb{P}^{\binom{n+d}{d}-1}$ as topological spaces.


Have I made an error in my reasoning, or am I right that the projective variety is $S/\sim$ instead of $S?$

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    $\begingroup$ The question is talking about the locus of the homogenous polynomials. And $f,g$ have the same locus iff $f \sim g$. So the exercise is correctly stated. $\endgroup$ Jan 22, 2018 at 23:02
  • $\begingroup$ It seems like you're trying to say that the parameter space of homogeneous polynomials of degree $d$ in $n + 1$ variables over $k$ is a projective variety. Is that correct? $\endgroup$ Jan 23, 2018 at 5:43
  • $\begingroup$ You mean the set of all homogeneous polynomials of that degree, not the locus. $\endgroup$ Jan 23, 2018 at 6:53

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Indeed, you need to projectivize the set of homogeneous polynomials of some degree for it to be a projective variety.

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