Smooth projectivevarieties with small invariants have received renewedinterest in recent years, primarily due to the fine study of the adjunction mapping. Now,through the effort of several mathematicians, a completeclassification of smooth surfaces in ${\Bbb P}^4$ has been worked out up todegree $10$, and a partial one is available in degree $11$.On the other side, recently Ellingsrud and Peskinehave provedHartshorne's conjecture that there are only finitely manyfamilies of smooth surfaces in ${\Bbb P}^4$, not of general type.It isbelieved that thedegree of the smooth, non-general type surfacesin ${\Bbb P}^4$ should be less than or equal to $15$.

The aim of this paper is to provide a series of examples ofsmooth surfaces in ${\Bbb P}^4$,not of general type, in degrees varying from $12$ up to $14$,and to describetheir geometry. By usingmainly syzygiesand 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.

A function f : Rn → R is a connectivity function if the graph of its restriction f|C to any connected C ⊂ Rn is connected in Rn x R. The main goal of this paper is to prove that every function f : Rn → R is a sum of n + 1 connectivity functions (Corollary 2.2). We will also show that if n > 1, then every function g: Rn → R which is a sum of n connectivity functions is continuous on some perfect set (see Theorem 2.5) which implies that the number n + 1 in our theorem is best possible (Corollary 2.6).

To prove the above results, we establish and then apply the following theorems which are of interest on their own.

For every dense Gδ-subset G of Rn there are homeomorphisms h1...,hn of Rn such that Rn=G∪h1(G)∪...∪hn(G) (Proposition 2.4).

For every n >1 and any connectivity function f : Rn → R, if x ∈ Rn and ε >0 then there exists an open set U⊂Rn such that x ∈ U⊂Bn(x,ε), f|bd(U) is continuous, and |(x)-f(y)| < ε for every y∈bd(U) (Proposition 2.7).