2022/02/01 by Kuroda, Shigeru
#14R20 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13A50 #Secondary 14R10
paper · doi:10.48550/arxiv.2202.00262
Let k be a field of characteristic p>0. We discuss the automorphisms of the polynomial ring k[x1,… ,xn] of order p, or equivalently the \bf Z/p\bf Z-actions on the affine space \bf Akn. When n=2, such an automorphism is know to be a conjugate of an automorphism fixing a variable. It is an open question whether the same holds when n≥ 3. In this paper, (1) we give the first counterexample to this question when n=3. In fact, we show that every \bf Ga-action on \bf Ak3 of rank three yields counterexamples for n=3. We give a family of counterexamples by constructing a family of rank three \bf Ga-actions on \bf Ak3. (2) For the automorphisms induced by this family of \bf Ga-actions, we show that the invariant ring is isomorphic to k[x1,x2,x3] if and only if the plinth ideal is principal, under some mild assumptions. (3) We study the Nagata type automorphisms of R[x1,x2], where R is a UFD of characteristic p>0. This type of automorphisms are of order p. We give a necessary and sufficient condition for the invariant ring to be isomorphic to R[x1,x2]. This condition is equivalent to the condition that the plinth ideal is principal.