Analysis of variance vs. homoscedasticity assumption If the assumption of homogeneity of variances is met,
how could the variances be different among groups in ANOVA
(the analysis dealing with between/within variability) ?
That is for me, ANOVA is something close to:
the ratio F = variance between groups / variance within groups.
So, if the (between/within) variances should be equal (according to the assumption),
F should be close to 1. Then what is the reason to have homoscedasticity assumption met before ANOVA (F test) calculation?

The assumption is that the variances within groups are equal. There is no assumption about the variances between groups. This means that the ratio variance between groups/variance within groups can take any value under the assumption.
(Qualification: The null hypothesis assumes that the true variance between group means is zero, so under the null hypothesis you'd expect the F-statistic to be not very large, but under the alternative it can become arbitrarily large, and in any case the equal variances within groups assumption alone doesn't constrain it in any way.)
