The Period and Index of a Generic Geometrically Elliptic Normal Curve
Eoin Mackall
TL;DR
This work addresses the problem of realizing and controlling the period and index invariants of genus one curves on Severi–Brauer varieties. It constructs a generic geometrically elliptic normal curve on the base extension of a generic Severi–Brauer variety with index $n$ and exponent $m$, proving $\mathrm{per}(C^{\mathrm{gen}}_{n,m})=n$ and $\mathrm{ind}(C^{\mathrm{gen}}_{n,m})=nm$ under the stated conditions, and shows that every admissible period/index pair arises by specialization. The approach rests on a versal construction via twisted Hilbert schemes $\mathbf{Hilb}^{\mathrm{tw}}_{\phi(t)}(\mathscr{X}/S)$, together with index-reduction arguments that pass from the generic curve to particular curves and use DVR-based specialization to establish sharp lower bounds. By bridging geometric-embedding data with Brauer-group and Picard-group calculations, the paper advances understanding of how period and index invariants can be engineered in families of genus one curves on Severi–Brauer varieties, with implications for division-algebra theory and arithmetic of curves.
Abstract
We construct genus one curves on base extensions of generic Severi--Brauer varieties of a given index and period which are versal objects for families of geometrically elliptic normal curves. We also compute the periods and indices of these curves showing that all possible period/index combinations are possible.