geometry ag.algebraica – A calculation of intersection homology

I am reading about perverse pulleys of the notes of Cataldo and Migliorini – 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.