Classification and geometric properties of surfaces with property ${\bf N}_{3,3}$

TL;DR

This paper classifies projective surfaces in P^5 satisfying property N_{3,3} using adjunction mappings, providing examples and exploring their CI-biliaison classes on cubic fourfolds.

Contribution

It offers the first classification of surfaces with property N_{3,3} in P^5, expanding understanding beyond the well-studied N_{2,e} cases.

Findings

Classification of surfaces with property N_{3,3} in P^5.

Examples of such surfaces constructed via Macaulay 2.

Analysis of CI-biliaison classes of degree 10 surfaces on cubic fourfolds.

Abstract

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.