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If NGC 6397 contains a large fraction of "second generation" stars (>70%according to recent analysis), the helium abundance of its stars might also beaffected, show some star-to-star variation, and be larger than the standard BigBang abundance Y~0.24. Can we derive constraints on this issue from theanalysis of the main sequence width and from its luminosity function? We buildup new models for the turnoff masses and the main sequence down to the hydrogenburning minimum mass, adopting two versions of an updated equation of state(EOS) including the OPAL EOS. Models consider different initial helium and CNOabundances to cover the range of possible variations between the first andsecond generation stars. We compare the models with the observational mainsequence. We also make simulations of the theoretical luminosity functions, fordifferent choices of the mass function and of the mixture of first and secondgeneration stars, and compare them with the observed luminosity function, bymeans of the Kolmogorov Smirnov --KS-- test. The study of the width of the mainsequence at different interval of magnitude is consistent with the hypothesisthat both generations are present in the cluster. If the CNO increase suggestedby spectroscopic observation is taken into account the small helium spread ofthe main sequence in NGC 6397 implies a substantial helium uniformity (DY~0.02)between first and second generation stars. The possible spread in heliumdoubles if an even larger increase of CNO is considered. The luminosityfunction is in any case well consistent with the observed data.

A detailed study of the main sequence of the Globular Cluster NGC 6397: can we derive constraints on the existence of multiple population?