The geometry of the Artin--Schreier-Mumford curves over an algebraically closed field

For a power q of a prime p, the Artin–Schreier–Mumford curve ASM(q) of genus g = (q − 1)2 is the nonsingular model X of the irreducible plane curve with affine equation (Xq + X)(Y q + Y ) = c, c ̸= 0, defined over a field K of characteristic p. The Artin–Schreier–Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for |c| < 1 they are curves with a large solvable automorphism group of order 2(q−1)q2 = 2√g(√g+1)2, far away from the Hurwitz bound 84(g−1) valid in zero characteristic; see [2–4]. In this paper we deal with the case where K is an algebraically closed field of characteristic p. We prove that the group Aut(X ) of all automorphisms of X fixing K elementwise has order 2q2(q − 1) and it is the semidirect product Q ⋊ Dq−1 where Q is an elementary abelian group of order q2 and Dq−1 is a dihedral group of order 2(q − 1). For the special case q = p, this result was proven by Valentini and Madan [13]; see also [1]. Furthermore, we show that ASM(q) has a nonsingular model Y in the three-dimensional projective space P G(3, K) which is neither classical nor Frobenius classical over the finite field Fq2