On absolute linear Harbourne constants

In the present note we study absolute linear Harbourne constants. These are invariants which were introduced in [2] in order to relate the lower bounds on the self-intersection of negative curves on birationally equivalent surfaces to the complexity of the birational map between them. We provide various lower and upper bounds on Harbourne constants and give their values for the number of lines s of the form p2r+pr+1 for any prime number p and also for all values of s up to 31. This extends considerably results of the third author obtained earlier in [10].