Derivation module and the Hilbert-Kunz multiplicity of the co-ordinate ring of a projective monomial curve

Let n1, n2, . . . , npbe a sequence of positive integers such thatn1< n2<? ? ?<npand gcd(n1, n2, . . . , np) = 1. LetS=?(0, np),(n1, np-n1), . . . ,(np-1, np?np-1),(np,0)?be an affine semigroup inN2. The semigroup ringk[S] is the co-ordinate ring of the projectivemonomial curve in the projective spacePpk, which is defined parametrically byx0=vnp, x1=un1vnp-n1, . . . , xp-1=unp-1vnp-np-1, xp=unp.We consider thatn1, n2, . . . , npforms an arithmetic sequence. Forp= 2,3, we give anexplicit set of minimal generators for the derivation module Derk(k[S]). Forp >3, wewrite an explicit formula for?(Derk(k[S])) and give a potential set of derivations for thederivation module Derk(k[S]). Further, we give an explicit formula for the Hilbert-Kunzmultiplicity of the co-ordinate ring of any projective monomial curve.