Forms of the affine line and their genus (Q1065091)

This paper studies forms of the affine line \({\mathbb{A}}^ 1\) over a field k in the following sense: K/k is inseparable of degree \(p=char(k)\), and X is a form of \({\mathbb{A}}^ 1\) if \(K\otimes X\cong {\mathbb{A}}^ 1_ K.\) Continuing work of Kambayashi, Miyanishi, and Takeuchi, who studied forms of arithmetical genus 0 and 1, we describe the numbers g which occur as genera of forms. Two wit, g occurs iff \(g=(p-1)\cdot j/2\) with \(j\neq - 1(p).\) \textit{P. Russell} [Pac. J. Math. 32, 527-539 (1970; Zbl 0199.245)] described by generators and relations those forms which carry a group structure. This is generalized to a wider class of forms called ''parallelizable''. X is parallelizable iff \(\Omega_ X\) is free cyclic as an \({\mathcal O}_ X\)-module. By considering the genus one sees that the parallelizable forms still constitute a small subclass of the class of all forms.