We present basic predictions of an updated version of the Munich semi-analytic hierarchical galaxy formation model that grows bulges via mergers and disk instabilities. Overall, we find that while spheroids below Ms ~ 10^11 Msun grow their sizes via a mixture of disk instability and mergers, galaxies above it mainly evolve via mergers. Including gas dissipation in major mergers, efficiently shrinks galaxies, especially those with final mass Ms < 10^11 Msun that are the most gas-rich, improving the match with different observables. We find that the predicted scatter in sizes at fixed stellar mass is still larger than the observed one by up to <40%. Spheroids are, on average, more compact at higher redshifts at fixed stellar mass, and at fixed redshift and stellar mass larger galaxies tend to be more starforming. More specifically, while for bulge-dominated galaxies the model envisages a nearly mass-independent decrease in sizes, the predicted size evolution for intermediate-mass galaxies is more complex. The z=2 progenitors of massive galaxies with mass around Ms and B/T>0.7 at z=0, are found to be mostly disc-dominated galaxies with a median B/T ~ 0.3, with only ~20% remaining bulge-dominated. The model also predicts that central spheroids living in more massive haloes tend to have larger sizes at fixed stellar mass. Including host halo mass dependence in computing velocity dispersions, allows the model to properly reproduce the correlations with stellar mass. We also discuss the fundamental plane, the correlations with galaxy age, the structural properties of pseudobulges, and the correlations with central black holes.
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