Stability of nilpotency of class 3 (Q1920761)

Let \(C_n\) be the scheme of all associative and commutative algebras of dimension \(n\) over a field \(K\) of characteristic \(\neq 2\). In this paper the authors consider the subscheme, say, \(A_n\) of those algebras that are nilpotent of class 3. They study the question when certain irreducible components of \(A_n\) are also components of \(C_n\). As was shown by \textit{I. R. Shafarevich} [Leningr. Math. J. 2, No. 6, 1335-1351 (1991); translation from Algebra Anal. 2, No. 6, 178-194 (1990; Zbl 0727.13006)] the components of \(A_n\) are given by \(A_{n,r}\) where the generic algebra \(N\) in \(A_{n,r}\) satisfies \(\dim N^2=r\). Moreover, \(A_{n,r}\) is a component of \(C_n\) if \(3\leq r\leq(d+1)(d+2)/6\). The authors prove the following two results: (1) If \(d\geq 9\) and \(d(d+1)/9\leq r\leq[d/3] (d-3)\), then there exists a point \(N\in A_{n,r}\) such that all regular tangent vectors to \(C_n\) at \(N\) are tangent to \(A_{n,r}\). By a regular tangent vector they mean, roughly speaking, that it occurs as tangent vector along a smooth curve in \(C_n\). (2) If \(d=4t>4\) and \(r=5t^2-2t\) then \(A_{n,r}\) is a component of \(C_n\).