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

Abstract

Let n0, n1,…,np be a sequence of positive integers such that n0<n1<⋯<np, gcd (n0,n1,… ,np)=1. Let S=⟨(0,np),(n0,np−n0),…,(np−1,np−np−1),(np,0)⟩ be an affine semigroup in N2. The semigroup ring k[S]is the coordinate ring of the projective monomial curve in the projective space Pp+1k, which is defined parametrically by x0=vnp,x1=un0vnp−n0,…,xp=unp−1vnp−np−1,xp+1=unp.

In this article, we consider the case when n0,n1,…,np forms an arithmetic sequence, and give an explicit set of minimal generators for the derivation module Der k(k[S]). Further, we give an explicit formula for the Hilbert–Kunz multiplicity of the coordinate ring of a projective monomial curve.