Hilbert-Kunz density function for graded domains

We prove the existence of HK density function for a graded pair ( R , I ) where is an N -graded domain finite type over perfect field and ⊂ ideal colength. This generalizes our earlier result one proves such where, in addition standard graded. Other properties functions also hold pairs: example, it multiplicative Segre products, its maximum support F -threshold m -primary provided Proj smooth, has closed formula when either generated by system parameters or dimension two. As consequences we show that if G group scheme acting linearly on polynomial ring d then f piecewise degree − 1 . compute S L 2 k [ X Y ]