On subcanonical surfaces of $Bbb Psp 4$.

We work over an algebraically closed field of characteristic zero. It is well known that the existence of indecomposable rank two vector bundles on P^n is equivalent, via the correspondance of Serre, to the existence of smooth X, of codimension two, which are subcanonical  and non complete intersections. If n is at least 4, basically, only one example of non split rank two vector bundle is known: the Horrocks-Mumford bundle on P^4. This bundle is associated to a smooth abelian surface of degree ten. This surface lies on a hyperquintic. In this paper we consider the problem of the existence of smooth subcanonical surfaces in P^4 lying on hypersurfaces of degree at most 4. If the degree of the hypersurface is 1 or 2, it is not diffcult to show that the surface is a complete intersection. Thanks to a result of Koelblen, the same conclusion holds true if the degree of the hypersurface is 3.