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The following proposition is an answer to Problem 2, and is also an answer to Problem 3 for cyclic algebras over a "big" field.

Proposition. For every Severi-Brauer $K$-variety $Y$ arising from a cyclic $K$-algebra of rank $n^2$, there exists a unique $\text{Aut}(Y)$-orbit of the Hilbert $K$-scheme of $Y$ parameterizing nodal, elliptic normal curves whose (geometric) irreducible components are lines, and this component has a $K$-point parameterizing such a curve $X_0.$ If $K$ is a "big" field in the sense of Florian Pop, e.g., the fraction field of a Henselian DVR, then there are also smooth elliptic normal curves $X$ in $Y$ obtained as deformations of $X_0.$ If the period of the cyclic algebra equals $n$, then every curve in $Y$ has index divisible by $n$.

If the field $K$ is "ample" or "big" in the sense of Florian Pop, then there are $K$-points of $U$. QED

Proof of the Corollary By Hensel's Lemma, the fraction field of every Henselian DVR is "big". For a local field $K$ such as $\mathbb{Q}_p$ or $\mathbb{F}_p((t)),$ every period-$n$ element in the Brauer group $\text{Br}(K)\cong \mathbb{Q}/\mathbb{Z}$ is represented by a cyclic $K$-algebra of rank $n^2$ by the Brauer-Hasse-Noether-(Albert) Theorem and Hasse's Structure Theorem. By the proposition, there exist $K$-points of $U$ parameterizing smooth elliptic normal curves $X$ in $Y$. QED

The following proposition is an answer to Problem 2, and is also an answer to Problem 3 for cyclic algebras over a "large" field.

Proposition. For every Severi-Brauer $K$-variety $Y$ arising from a cyclic $K$-algebra of rank $n^2$, there exists a unique $\text{Aut}(Y)$-orbit of the Hilbert $K$-scheme of $Y$ parameterizing nodal, elliptic normal curves whose (geometric) irreducible components are lines, and this component has a $K$-point parameterizing such a curve $X_0.$ If $K$ is a "large" field in the sense of Florian Pop, e.g., the fraction field of a Henselian DVR, then there are also smooth elliptic normal curves $X$ in $Y$ obtained as deformations of $X_0.$ If the period of the cyclic algebra equals $n$, then every curve in $Y$ has index divisible by $n$.

If the field $K$ is "ample" or "large" in the sense of Florian Pop, then there are $K$-points of $U$. QED

Proof of the Corollary By Hensel's Lemma, the fraction field of every Henselian DVR is "large". For a local field $K$ such as $\mathbb{Q}_p$ or $\mathbb{F}_p((t)),$ every period-$n$ element in the Brauer group $\text{Br}(K)\cong \mathbb{Q}/\mathbb{Z}$ is represented by a cyclic $K$-algebra of rank $n^2$ by the Brauer-Hasse-Noether-(Albert) Theorem and Hasse's Structure Theorem. By the proposition, there exist $K$-points of $U$ parameterizing smooth elliptic normal curves $X$ in $Y$. QED

There are positive results for general Severi-Brauer varieties and general fields for small values of the integer $n$. One such result was reported by David Saltman at a seminar in Fall 2016 at Stony Brook University:  https://www.math.stonybrook.edu/deptcalendar/event.php?ID=3964&Date=2016-11-02

There are positive results for general Severi-Brauer varieties and general fields for small values of the integer $n$. One such result was reported by David Saltman at a seminar in Fall 2016 at Stony Brook University: 

https://www.math.stonybrook.edu/deptcalendar/event.php?ID=3964&Date=2016-11-02

For some reason, the link to the Stony Brook University calendar is demanding a password(!), so here is a link to a seminar announcement for a similar seminar by David Saltman at NYU.

https://math.nyu.edu/dynamic/calendars/seminars/algebraic-geometry-seminar/846/

Open Problem. For a Severi-Brauer variety $Y$ over a field $K$, i.e., a $K$-scheme $Y$ such that $Y\times_{\text{Spec}\ K}\text{Spec}\ \overline{K}$ is $\overline{K}$-isomorphic to $\mathbb{P}^{n-1}_{\overline{K}},$ does there exist a genus $1$ $K$-curve $X$ and a $K$-morphism $X\to Y?$

Open Problem. For every Severi-Brauer variety $Y$ over a field $K$, i.e., for every $K$-scheme $Y$ such that $Y\times_{\text{Spec}\ K}\text{Spec}\ \overline{K}$ is $\overline{K}$-isomorphic to $\mathbb{P}^{n-1}_{\overline{K}},$ does there exist a genus $1$ $K$-curve $X$ and a $K$-morphism $X\to Y?$