General position on Severi--Brauer surfaces

The blowing-up of the projective plane at a finite set of points yields a del Pezzo surface if and only if the points lie in general position. In this note, we generalize this result to Severi–Brauer surfaces over arbitrary ground fields. Using Galois descent, intersection theory and combinatorial arguments, we provide explicit arithmetic and geometric conditions on the centre of the blowing-up.