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Let $d>0$ and $N=\binom{n+d}{n}-1$. Then $\mathbb{P}^N$ is the parameter space of degree $d$ homogeneous polynomials in $n+1$ variables $x_0,\dots,x_n$ over an algebraically closed field $k$.

Let $\Gamma\subset \mathbb{P}^N$ be the subset whose elements correspond to irreducible degree $d$ homogeneous polynomials. Is $\Gamma$ an irreducible algebraic set?

rpf
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  • No, it's not even closed, it's open: https://math.stackexchange.com/questions/459868/are-most-polynomials-reducible-or-irreducible?rq=1 – Qiaochu Yuan Oct 11 '17 at 03:23

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