서강대학교

초록/요약

A numerical Godeaux surface X is a minimal surface of general type with \chi(\Oh_{X})=K_{X}^{2}=1. In this thesis we abbreviate numerical Godeaux surfaces to Godeaux surfaces. Godeaux constructed complex surfaces of general type with the smallest invariants p_g=q=0 and K_X^2=1 as the quotient of a smooth quintic surface in \PP^3 by a free \ZZ/5\ZZ action \cite{G31}. Godeaux surfaces with an involution over \CC were studied by Keum and Lee \cite{KL00}, and subsequently Calabri, Ciliberto and Mendes Lopes \cite{CCMP} classified the possible cases for the quotient space X/ \sigma by its involution \sigma. They proved that the quotient surface is either rational or birational to an Enriques surface.The facts that \Tors X has order at most 5 and \ZZ/2\ZZ \oplus\ZZ/2\ZZ is impossible were proved by Bombieri, Miyaoka, and Reid \cite{Bom73}, \cite{Mi76}, \cite{Re78}. On the other hand, a minimal surface X of general type with K_X^2=1 and \chi(\Oh_X)=1 can have p_g(X)=h^1(\Oh_X)=1 in characteristic p=2,3 and 5 \cite{Lie09}. These Godeaux surfaces have nonreduced \Pic X. Lang \cite{La81} showed Godeaux surfaces exist in every characteristic. We study the Godeaux surfaces constructed by original Godeaux method in characteristic 5 due to Lang \cite{La81}, Miranda \cite{Mi84}, and Liedtke \cite{Lie09} with \PictauX \iso\ZZ/5\ZZ and infinitesimal group schemes \bmu_5 or \bal_5 respectively. We work over an algebraically closed field in odd characteristic. We prove |\Tors X |\leq 5 in characteristic p >3. We show that the quotient surface X/ \sigma by its involution is rational, or is birational to an Enriques surface through careful study of the Kawamata-Viehweg vanishing theorem and numerical calculations in positive characteristic. However, we prove that the Enriques case cannot occur in the cases of nonreduced \Pic X in characteristic 3 or 5, and hence X/\sigma must be rational in these cases. In addition we construct explicit examples in characteristic 5 of quintic hypersurfaces Y having an action of each of the group schemes G of order 5 with extra symmetry by \Aut G \cong \ZZ/4\ZZ. In these three examples X = Y/G is a nonsingular Godeaux surface with an action of \ZZ/4\ZZ, and in particular have an involution \sigma. The G action on the quintic hypersurface extends to an action of the {\em holomorph} H_{20} of G, that contains a group scheme of order 10 analogous to the dihedral group D_5. For the two infinitesimal cases G=\bmu_5 and \bal_5, the nonsingularity of X involves a nonclassical calculation instead of Bertini's theorem. In both cases, we prove that a general Y has exactly 11 singular points of type A_4.