index of smooth varieties

Question

What are the simplest examples of smooth, projective varieties defined over the fraction field of an Henselian DVR of characteristic $0$ which have index $>1$?

EDIT: Also assume that the residue field of the DVR is algebraically closed.

Answer 1

Let $d\geq 3$ be any integer that is relatively prime to the residue characteristic of $R$. Let $s$ be any generator of the maximal ideal $\mathfrak{m}$ of $R$. Let $\widetilde{R}$ be the finite, flat extension $R[\sigma]/\langle \sigma^d -s \rangle$ with its natural action of the group scheme $\mu_d=\text{Spec} \ \mathbb{Z}[\zeta]/\langle \zeta^d-1\rangle$, $$\text{Spec}\ \mathbb{Z}[\zeta]\langle \zeta^d-1 \rangle \times \text{Spec}\ R[\sigma]/\langle \sigma^d-s\rangle \to \text{Spec}\ R[\sigma]/\langle \sigma^d - s\rangle, \ \ \sigma \mapsto \zeta\otimes \sigma.$$ Denote the induced morphism of schemes by $$q:\text{Spec}\ \widetilde{R} \to \text{Spec}\ R.$$ The generic fiber of $q$ is a $\mu_d$-torsor.

Let $X_R$ be the smooth, projective $R$-scheme, $$X_R = \text{Proj}\ R[t_0,t_1,\dots,t_{d-1}]/\langle t_0^d + t_1^d + \dots + t_{d-1}^d \rangle.$$ This has a natural action of $\mu_d$ by $$\zeta\bullet [t_0,t_1,\dots,t_{d-1}] = [\zeta^0t_0,\zeta^1 t_1, \dots, \zeta^{d-1}t_{d-1}].$$ Denote by $X_{\widetilde{R}}$ the base change, $$X_{\widetilde{R}} = \text{Spec}\ \widetilde{R} \times_{\text{Spec}\ R} X_R.$$ This has a diagonal action of $\mu_d$ that is a free action. The projection to the first factor is equivariant for this action, $$\text{pr}_1:X_{\widetilde{R}} \to \text{Spec}\ \widetilde{R}.$$ Thus, for the geometric quotient by this free $\mu_d$-action, $$q_X:X_{\widetilde{R}} \to \mathcal{X}_R^s,$$ there is a unique morphism, $$\pi:\widetilde{X}_R\to \text{Spec}\ R,$$ such that $\pi\circ q_X$ equals $q\circ \text{pr}_1$. Since the base change of $\pi$ by the fppf morphism $q$ equals the smooth morphism $\text{pr}_1$, also the proper morphism $\pi$ is smooth. Since the geometric generic fiber of $\text{pr}_1$ is a smooth hypersurface of dimension $d-2$, it is integral. Thus, the geometric generic fiber of $\pi$ is also integral.

If $d$ is a prime, then the index of the generic fiber of $\pi$ equals $d$. Indeed, every zero-dimensional, reduced, closed subscheme of the generic fiber pulls back to a zero-dimensional, reduced, closed subscheme of the generic fiber of $\text{pr}_1$ that is $\mu_d$-invariant. Thus, for every irreducible component of this closed subscheme, the closure in $X_{\widetilde{R}}$ is an integral, closed subscheme that is $\mu_d$-invariant and finite over $\text{Spec}\ \widetilde{R}$. By the valuative criterion of properness for $\text{pr}_1$, the intersection of this closed subscheme with the closed fiber of $\text{pr}_1$ is a closed subscheme that is $\mu_d$-invariant. Since the action of $\mu_d$ on the closed fiber is free, the length is divisible by $d$. Since this holds for every irreducible component, the total length is divisible by $d$.