  The type II_1 unprojection is, by definition, the generic complete
intersection type II unprojection, in the sense of [Papadakis, Type II
unprojection, J. Algebraic Geometry, 15 (2006) 399--414] Section 3.1, for the
parameter value k = 1, and depends on a parameter n greater or equal than 2.
Our main results are the explicit calculation of the linear relations of the
type II_1 unprojection for any value of n greater or equal than 2 (Theorem
3.16) and the explicit calculation of the quadratic equation for the case n = 3
(Theorem 4.1). In addition, Section 5 contains applications to algebraic
geometry while Section 6 contains the Macaulay 2 code for the type II_1
unprojection for the parameter value n = 3.
