The cross sections for the photofission of plutonium isotopes [sup 238]Pu, [sup 240]Pu, and [sup 242]Pu in the energy range 5-10 MeV have been measured by using a beam of bremsstrahlung photons from the microtron installed at the Institute of Physics and Power Engineering (Obninsk). The energy regions below the fission barrier and above 6 MeV have been scanned with pitches of 0.05 and 0.1 MeV, respectively. In deriving the absolute cross section for [sup 238]Pu photofission, [sup 238]U photofission has been employed as a reference reaction. In measurements involving [sup 240]Pu and [sup 242]Pu nuclei, the neptunium isotope [sup 237]Np, which is characterized by a more regular dependence of the photofission cross section on excitation energy than [sup 238]U, has been chosen for the first time as a reference nucleus. The measured cross sections as functions of energy show resonance structures in the vicinity of the fission threshold that are consistent with those previously observed in the energy dependences of fissilities for corresponding direct reactions. The partial-wave cross sections for the J[sup π]K = 1[sup -]0, 1[sup -]1, and 2[sup +]0 photofission channels have been determined as functions of energy. At energies below some 5.5 MeV, the total cross section for photofission of each plutonium isotope being studied receives a significant contribution from quadrupole interaction. Within the one-dimensional model of a two-humped fission barrier, the parameters of the barriers for [sup 238]Pu, [sup 240]Pu, and [sup 242]Pu have been extracted from data and have then been compared with estimates based on previous measurements. [ABSTRACT FROM AUTHOR]