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.



index of smooth varieties
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 $d$.


