The ESO Key-Programme ``A Homogeneous Bright QSO Survey'' - I The Methods and the ``Deep'' Fields

This is the first paper in a series aimed at defining a statistically significant sample of QSOs in the range $ 15 < B < 18.75$ and $ 0.3 < z < 2.2$. The selection is carried out using direct plates obtained at the ESO and UK Schmidt Telescopes, scanned with the COSMOS facility and searched for objects with an ultraviolet excess. Follow-up spectroscopy, carried out at ESO La Silla, is used to classify each candidate. In this initial paper, we describe the scientific objectives of the survey; the selection and observing techniques used. We present the first sample of 285 QSOs ($M_B < -23$) in a 153 deg$^2$ area, covered by the six ``deep'' fields, intended to obtain significant statistics down $B \simeq 18.75$ with unprecedented photometric accuracy. From this database, QSO counts are determined in the magnitude range $ 17 < B < 18.75$