On general surfaces defined by an almost linear pfaffian

Abstract

We calculate the integers d such that a general surface Xd in \({\mathbb{P}^{3}}\) of degree d contains an arithmetically Gorenstein set of points with a linear syzygy matrix of size 2α + 1. This condition is equivalent to Xd being defined by the pfaffian of a skew-symmetric matrix whose entries are linear except possibly a row and a column. We prove that this takes place for all d ≥ α + 1 if α ≤ 10. Conversely, for α ≥ 11, we show that the condition holds if and only if d is contained in the interval [α + 1, 15].