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.
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.
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 $d$.