Asymptotic N P Property of Rational Surfaces

The pioneering work of Mumford ([M]), its amplifications by St. Donat ([SD]) and Fujita ([F]), the inspiring work of Green ([Gr]), followed by works of Green and Lasarsfeld ([G-L]) and Ein and Lasarsfeld ([E-L]), have captured the interest and have influenced a large number of researchers in the last fifteen years. Several authors have studied the defining equations of projective varieties and, more generally, the higher order syzygies among these equations. A significant algebraic property was introduced along this line of works ([Gr], [G-L]), the N p property. It says that a variety is generated by quadratics, and its minimal free resolution is linear up to the first p steps. We recall in detail the definition of N p property from [G-L].