  Let $X$ be a closed subscheme of codimension $e$ in a projective space. One
says that $X$ satisfies property ${\bf N}_{d,p}$, if the $i$-th syzygies of the
homogeneous coordinate ring are generated by elements of degree $<d+i$ for
$0\le i\le p$. The geometric and algebraic properties of smooth projective
varieties satisfying property ${\bf N}_{2,e}$ are well understood, and the
complete classification of these varieties is a classical result. The aim of
this paper is to study the next case: projective surfaces in $\Bbb P^5$
satisfying property ${\bf N}_{3,3}$. In particular, we give a classification of
such varieties using adjunction mappings and we also provide illuminating
examples of our results via calculations done with Macaulay 2. As corollaries,
we study the CI-biliaison equivalence class of smooth projective surfaces of
degree $10$ satisfying property ${\bf N}_{3,3}$ on a cubic fourfold.
