Geodesic
The moduli space of commutative algebras of finite rank
Journal of the European Mathematical Society, Tome 10 (2008) no. 3, pp. 817-836 Cet article a éte moissonné depuis la source EMS Press

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The moduli space of rank-n commutative algebras equipped with an ordered basis is an affine scheme Bn​ of finite type over Z, with geometrically connected fibers. It is smooth if and only if n≤3. It is reducible if n≥8 (and the converse holds, at least if we remove the fibers above 2 and 3). The relative dimension of Bn​ is 272​n3+O(n8/3). The subscheme parameterizing étale algebras is isomorphic to GLn​/Sn​, which is of dimension only n2. For n≥8, there exist algebras that are not limits of étale algebras. The dimension calculations lead also to new asymptotic formulas for the number of commutative rings of order pn and the dimension of the Hilbert scheme of n points in d-space for d≥n/2.
DOI : 10.4171/jems/131
Classification : 14-XX, 13-XX, 00-XX 
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     author = {Bjorn Poonen},
     title = {The moduli space of commutative algebras of finite rank},
     journal = {Journal of the European Mathematical Society},
     pages = {817--836},
     year = {2008},
     volume = {10},
     number = {3},
     doi = {10.4171/jems/131},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/131/}
}
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Bjorn Poonen. The moduli space of commutative algebras of finite rank. Journal of the European Mathematical Society, Tome 10 (2008) no. 3, pp. 817-836. doi: 10.4171/jems/131

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