Examples of Smooth Non-General Type Surfaces in P4

Abstract

Smooth projective varieties with small invariants have received renewed interest in recent years, primarily due to the fine study of the adjunction mapping. Now, through the effort of several mathematicians, a complete classification of smooth surfaces in ${\Bbb P}^4$ has been worked out up to degree $10$, and a partial one is available in degree $11$. On the other side, recently Ellingsrud and Peskine have proved Hartshorne's conjecture that there are only finitely many families of smooth surfaces in ${\Bbb P}^4$, not of general type. It is believed that the degree of the smooth, non-general type surfaces in ${\Bbb P}^4$ should be less than or equal to $15$. The aim of this paper is to provide a series of examples of smooth surfaces in ${\Bbb P}^4$, not of general type, in degrees varying from $12$ up to $14$, and to describe their geometry. By using mainly syzygies and liaison techniques, we construct the following families of surfaces: \begin{enumerate} \item[] minimal proper elliptic surfaces of degree $12$ and sectional genus $\pi=13$; \item[] two types of non-minimal proper elliptic surfaces of degree $12$ and sectional genus $\pi=14$; \item[] non-minimal $K3$ surfaces of degree $13$ and sectional genus $16$; and \item[] non-minimal $K3$ surfaces of degree $14$ and sectional genus $19$. 1991 Mathematics Subject Classification: 14M07, 14J25, 14J26, 14J28, 14C05.