On syzygies of linear sections

In this paper, we study how the minimal free resolution of a closed subscheme X ⊂ P r X \subset \mathbb {P}^r relates to that of a linear section X ∩ Λ ⊂ Λ = P s X \cap \Lambda \subset \Lambda = \mathbb {P}^s ( 0 > s > r ) (0 > s >r) . Our main result implies that the shape of the final non-zero row of the Betti diagram of X X is preserved under taking the zero-dimensional and one-dimensional linear sections.