Hironaka's additive group scheme. II (Q1061187)

Hironaka's additive group schemes are subgroup schemes of a vector group scheme over a field k of positive characteristic p. In part I of this paper [in Number Theory, Algebr. Geom., Commut. Algebra, in Honor of Y. Akizuki, 181-219 (1973; Zbl 0287.14014)] the author showed that these group schemes had a ''linear-algebra-type'' description. In the present work, he provides an alternative description which in addition allows him to produce a versal family of Hironaka subgroup schemes. - The description, while explicit, is difficult to summarize. Briefly: let S be a polynomial ring over k, \(L_ 0\) the k-subspace of linear forms in S, K the field of all elements such that some iterated Frobenius takes them to k and \(K_ e\) whose \(p^ e\)-th power is in k. V is a subspace of \(K\otimes L_ 0\) satisfying a technical condition. Let L(B) be the sum of the \(p^ e\)-th powers of \(V\cap (K_ e\otimes L_ 0)\), let I be the ideal that it generates in S, and let \(B=Spec(S/I)\). Then B is a Hironaka subgroup scheme of the vector group Spec(S), and all such arise this way.