Context. A subclass of broad extreme ultraviolet (EUV) and X-ray jets, called blowout jets, have become a topic of research since they could be the link between standard collimated jets and coronal mass ejections (CMEs). Aims: Our aim is to understand the origin of a series of broad jets, some of which are accompanied by flares and associated with narrow and jet-like CMEs. Methods: We analyze observations of a series of recurrent broad jets observed in AR 10484 on 21-24 October 2003. In particular, one of them occurred simultaneously with an M2.4 flare on 23 October at 02:41 UT (SOLA2003-10-23). Both events were observed by the ARIES Hα Solar Tower-Telescope, TRACE, SOHO, and RHESSI instruments. The flare was very impulsive and followed by a narrow CME. A local force-free model of AR 10484 is the basis to compute its topology. We find bald patches (BPs) at the flare site. This BP topology is present for at least two days before to events. Large-scale field lines, associated with the BPs, represent open loops. This is confirmed by a global potential free source surface (PFSS) model. Following the brightest leading edge of the Hα and EUV jet emission, we can temporarily associate these emissions with a narrow CME. Results: Considering their characteristics, the observed broad jets appear to be of the blowout class. As the most plausible scenario, we propose that magnetic reconnection could occur at the BP separatrices forced by the destabilization of a continuously reformed flux rope underlying them. The reconnection process could bring the cool flux-rope material into the reconnected open field lines driving the series of recurrent blowout jets and accompanying CMEs. Conclusions: Based on a model of the coronal field, we compute the AR 10484 topology at the location where flaring and blowout jets occurred from 21 to 24 October 2003. This topology can consistently explain the origin of these events. The movie associated to Fig. 1 is available at http://www.aanda.org