Context. Ultraviolet (UV) emission-line spectra are used to spectroscopically confirm high-z galaxies and increasingly also to determine their physical properties.

Aims. We construct photoionization models to interpret the observed UV spectra of distant galaxies in terms of the dominant radiation field and the physical condition of the interstellar medium (ISM). These models are applied to new spectroscopic observations from the VIMOS Ultra Deep Survey (VUDS).

Methods. We construct a large grid of photoionization models, which use several incident radiation fields (stellar populations, active galactic nuclei (AGNs), mix of stars and AGNs, blackbodies, and others), and cover a wide range of metallicities and ionization parameters. From these models we derive new spectral UV line diagnostics using equivalent widths (EWs) of [CIII]lambda 1909 doublet, CIV lambda 1549 doublet and the line ratios of [CIII], CIV, and He II lambda 1640 recombination lines. We apply these diagnostics to a sample of 450 [CIII]-emitting galaxies at redshifts z = 2-4 previously identified in VUDS.

Results. We demonstrate that our photoionization models successfully reproduce observations of nearby and high-redshift sources with known radiation field and/or metallicity. For star-forming galaxies our models predict that [CIII] EW peaks at sub-solar metallicities, whereas CIV EW peaks at even lower metallicity. Using the UV diagnostics, we show that the average star-forming galaxy (EW([CIII]) similar to 2 angstrom) based on the composite of the 450 UV-selected galaxies' spectra The inferred metallicity and ionization parameter is typically Z = 0.3-0.5 Z(circle dot) and logU = 2.7 to 3, in agreement with earlier works at similar redshifts. The models also indicate an average age of 50-200 Myr since the beginning of the current star-formation, and an ionizing photon production rate, xi(ion), of log xi(ion)/erg(-1) Hz = 25.3-25.4. Among the sources with EW([CIII]) > = 10 angstrom, approximately 30% are likely dominated by AGNs. The metallicity derived for galaxies with EW(CIII) = 10-20 angstrom is low, Z = 0.02-0.2 Z(circle dot), and the ionization parameter higher (logU similar to -1.7) than the average star-forming galaxy. To explain the average UV observations of the strongest but rarest [CIII] emitters (EW([CIII]) > 20 angstrom), we find that stellar photoionization is clearly insufficient. A radiation field consisting of a mix of a young stellar population (log xi(ion.)/ erg(-1) Hz similar to 25.7) plus an AGN component is required. Furthermore an enhanced C/O abundance ratio (up to the solar value) is needed for metallicities Z = 0.1-0.2 Z(circle dot) and logU = -1.7 to -1.5.

Conclusions. A large grid of photoionization models has allowed us to propose new diagnostic diagrams to classify the nature of the ionizing radiation field (star formation or AGN) of distant galaxies using UV emission lines, and to constrain their ISM properties. We have applied this grid to a sample of [CIII]-emitting galaxies at z = 2-4 detected in VUDS, finding a range of physical properties and clear evidence for significant AGN contribution in rare sources with very strong [CIII] emission. The UV diagnostics we propose should also serve as an important basis for the interpretation of upcoming observations of high-redshift galaxies.