On normal quintic Enriques surfaces.

Abstract

In this thesis, we are concerned with two families of normal quintic surfaces in P$\sp3$ constructed by Ezio Stagnaro, which we call normal quintic surfaces of the first and second kind in P$\sp3$. The aim of this thesis is to reproduce Ezio Stagnaro's claim that these two families of normal quintic surfaces are birationally isomorphic to Enriques surfaces using modern language. Moreover we characterize each of these two birational models by very special classes of divisor and find their moduli numbers. One of our main results is that every Enriques surface is birationally isomorphic to a normal surface in P$\sp3$. For the first model, we first find a divisor class on the Enriques surfaces which correspond to hyperplane sections of normal quintic surfaces of the first kind by birational isomorphisms, and show that most of Enriques surfaces with this special type of divisors are birationally isomorphic to normal quintic surfaces of the first kind. After showing that four tacnodal singular points give linearly independent conditions on the space of quintic surfaces in P$\sp3$, we count the moduli number of Enriques surfaces obtained from normal quintic surfaces of the first kind. Enriques surfaces from normal quintic surfaces of the first kind are also constructed from the etale double covering K3 surfaces which are the base spaces of nets of quadrics in P$\sp5$. We characterize nets of quadrics from which we could get Enriques surfaces with the special class of divisors obtained from normal quintic surfaces of the first kind. Then we count the moduli of these special nets of quadrics and describe their Hessian curves. Next we discuss normal quintic surfaces of the second kind in P$\sp3$. Similarly to the first model, we characterize Enriques surfaces obtained from normal quintic surfaces of the second kind by a special type of divisors, and then show that every Enriques surface has this special type of divisor on it, thereby we get the previously stated result. We also show that K3 surfaces which are the etale double coverings of Enriques surfaces are birationally isomorphic to normal quintic surfaces in P$\sp3$.