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Abstract
Let X be a closed subscheme of codimension e in a projective space. One says that X satisfies property Nd,p, if the i-th syzygies of the homogeneous coordinate ring are generated by elements of degree <d+i for 0≤i≤p. The geometric and algebraic properties of smooth projective varieties satisfying property N2,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 P5 satisfying property N3,3. 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 N3,3 on a cubic fourfold.
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