I am reading about perverse pulleys of the notes of Cataldo and Migliorini http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260 – 9.pdf

On page 553 of example 2.2.2 they say:

Yes $ Y $ It is the projective cone on a non-singular curve. $ C $ of the genre $ G $ then the cohomology groups are $ mathbb {Q}, 0, mathbb {Q}, mathbb {Q} ^ {2g}, mathbb {Q} $, while intersectional cohomology groups are $ mathbb {Q}, mathbb {Q} ^ {2g}, mathbb {Q}, mathbb {Q} ^ {2g}, mathbb {Q}. $

Can someone explicitly explain these two calculations and perhaps other basic calculations of the intersection of cohomology? For some reason, it seems that I can not understand why this is so.