The geometric classification of nilpotent commutative $\mathfrak{CD}$-algebras

We give a geometric classification of complex $5$-dimensional nilpotent commutative $\mathfrak{CD}$-algebras. The corresponding geometric variety has dimension $24$ and decomposes into $10$ irreducible components determined by the Zariski closures of a two-parameter family of algebras, three one-parameter families of algebras, and $6$ rigid algebras.

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