International audienceWe study the pure braid groups Pn(RP2) of the real projective plane RP2, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1→Pm(RP2x1,...,xn→Pn+m(RP2)→p∗Pn(RP2)→1, where n≥2 and m≥1, and p∗ is the homomorphism which corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration p:Fn+m(RP2)→Fn(RP2) of configuration spaces. Van Buskirk proved in 1966 that p and p∗ admit a section if n=2 and m=1. Our main result in this paper is to prove that there is no section if n≥3. As a corollary, it follows that n=2 and m=1 are the only values for which a section exists. As part of the proof, we derive a presentation of Pn(RP2): this appears to be the first time that such a presentation has been given in the literature