Derivation module and the Hilbert-Kunz multiplicity of the coordinate ring of a projective monomial curve (Q6595531)

This paper describes several invariants of a specific type of affine semigroup ring: the coordinate rings of monomial curves in projective space. More precisely, say \(S\) is the semigroup in \(\mathbb N^2\) generated by \((0,n_p), (n_0,n_p-n_0),(n_1,n_p-n_1),\dots,(n_p,0)\), with \(n_0<\dots<n_p\) sharing no common divisor, and \(k[S]\) the resulting semigroup algebra. In this setting, \(k[S]\) is the homogeneous coordinate ring of the parametrized curve in \(\mathbb P^{p+1}\) defined by \(x_0=v^{n_p},x_1=u^{n_0}v^{n_p-n_0},\dots,x_{p+1}=u^{n_p}\). In general, it is of interest to understand the structure of the module of differentials \(\mathrm{Der}_k(k[S])\) (dual to the Kahler differentials \(\Omega_{k[S]/k}\)), whose properties reflects the singularities of \(k[S]\).\N\NUnder the additional assumption that the \(n_0,\dots,n_p\) form a minimal arithmetic sequence, this paper gives explicit minimal generators for the derivation module \(\mathrm{Der}_k(k[S])\). (In particular, they calculate the minimal number of generators.) In particular, this gives the minimal generators of \(\mathrm{Der}_k(k[S])\) for projective monomial curve in \(\mathbb P^2\) corresponding to the semigroup generated by \((0,n_1)\), \((n_0,n_1-n_0)\), \((n_1,0)\). Furthermore, the paper also describes the Hilbert--Kunz multiplicity \(e_{\mathrm{HK}}(k[S]_{ m})\) when the \(a_i\) do not necessarily form a minimal arithmetic progression. The Hilbert--Kunz multiplicity of a local ring \((R,m)\) is an invariant of singularities in characteristic \(p\) that describes the asymptotic behavior of the length of \(R/m^{[p^e]}\) as \(e \to \infty\), and is in general quite hard to compute. The resulting formula is combinatorial and does not depend on the characteristic \(p\).