Nonclassical Godeaux surfaces in characteristic five

Abstract:

A classical Godeaux surface is a smooth minimal projective surface $X$, with $K_X^2 = 1$, ${p_a} = {p_g} = 0$ and ${\text {Pi}}{{\text {c}}^\tau }(X) = {\mathbf {Z}}/5{\mathbf {Z}}$. A nonclassical Godeaux surface is a smooth minimal projective surface $X$ with $K_X^2 = 1$, ${p_a} = 0$, ${p_g} = 1$ and ${\text {Pi}}{{\text {c}}^\tau }(X) = {\mu _5}$ or ${\alpha _5}$; such surfaces should exist in characteristic 5. It is the purpose of this note to construct nonclassical Godeaux surfaces in characteristic 5, with ${\text {Pi}}{{\text {c}}^\tau }(X) = {\mu _5}$. The method is to exhibit a smooth quintic surface on which ${\mathbf {Z}}/5{\mathbf {Z}}$ acts, so that the quotient is smooth; this quotient is the desired surface.