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Let $n\geq 1$ be an integer. There is an obvious family of $n$-dimensional unital algebras parametrized by $\mathbb{C}^{n(n-1)^2}$ such that any $n$-dimensional unital algebra is isomorphic to at least one algebra in the family.

A closed (possibly reducible) subvariety $V\subset \mathbb{C}^{n(n-1)^2}$ is representative if any $n$-dimensional commutative associative unital algebra is isomorphic to at least one algebra in the corresponding subfamily.

For $n\leq 6$ there is a zero-dimensional representative subvariety. What can we say about the minimum dimension of a representative subvariety for $n\geq 7$?

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    $\begingroup$ I don’t understand your second paragraph but see: arxiv.org/abs/math/0608491 $\endgroup$
    – Qiaochu Yuan
    Commented Oct 12, 2020 at 17:40
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    $\begingroup$ see also this answer and comments there. In particular, if $m_n$ is this minimal dimension, then $\liminf m_n/n^3>0$ (with an explicit lower bound I'm just lazy to retrieve). $\endgroup$
    – YCor
    Commented Oct 13, 2020 at 9:10

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