Consider the Hilbert scheme of degree $n$, genus $1$ curves in $\mathbb P^{n-1}$. It contains the locus of smooth curves embedded by the complete linear system of a degree $n$ divisor. Let $X_n$ be the closure of this locus. What is the geometry of $X_n$?
For $n=3$, $X_n = \mathbb P^9$ with each curve the vanishing set of a cubic polynomial in three variables.
For $n=4$, $X_n=Gr(2,10)$ with each curve the vanishing set of a two-dimensional subspace of the ten-dimensional space of quadratic polynomials in four variables.
For $n=5$, $X_n=V/GL_5$ where $V$ is the sum of five copies of the $\wedge^2$ representation of $GL_5$, by a weird explicit isomorphism.
In general $X_n$ has dimension $n^2$.
The open subscheme of $X_n$ parameterizing smooth curves is equal to a bundle over the moduli space of elliptic curves with fiber $PGL_n / (\mathbb Z/n \times \mathbb Z/n)$, where the $(\mathbb Z/n \times \mathbb Z/n)$ subgroup of $PGL_n$ is the group of translations and modulations. Hence $X_n$ is always unirational.
What else can we say about the geometry of $X_n$? Is it always rational? Is it always smooth? Fano? Is it always the quotient of an affine space by a group action?
The explicit descriptions of $X_n$ for $n=3,4,5$ were used by Manjul Bhargava and Arul Shankar to count rational points on $X_n$ and thereby to compute the average size of the $n$-Selmer groups of elliptic curves for $n=3,4,5$. If $X_n$ for any higher $n$ was known to be have enough nice properties, it would be possible to do the same thing, and then to get new results on average ranks of elliptic curves.
