DERIVATION MODULE AND THE HILBERT–KUNZ MULTIPLICITY OF THE COORDINATE RING OF A PROJECTIVE MONOMIAL CURVE

Abstract

Let $n_0,n_1,\dots ,n_p$ be a sequence of positive integers such that , $\text{gcd }(n_0,n_1,\text{… },n_p)=1$. Let $S=⟨(0,n_p),(n_0,n_p-n_0),\dots ,(n_{p-1},n_p-n_{p-1}),(n_p,0)⟩$ be an affine semigroup in ${\mathbb{N}}^2$. The semigroup ring $k[S]$ is the coordinate ring of the projective monomial curve in the projective space ${\mathbb{P}}_k^{p+1}$, which is defined parametrically by

$$x_0=v^{n_p},x_1=u^{n_0}{v}^{n_p-n_0},\dots ,x_p=u^{n_{p-1}}{v}^{n_p-n_{p-1}},x_{p+1}=u^{n_p}.$$

In this article, we consider the case when $n_0,n_1,\dots ,n_p$ forms an arithmetic sequence, and give an explicit set of minimal generators for the derivation module ${\text{Der }}_k(k[S])$. Further, we give an explicit formula for the Hilbert–Kunz multiplicity of the coordinate ring of a projective monomial curve.