Hilbert scheme of projectively normal elliptic curves

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.

• For $n = 6$, I guess, one can use the fact, that an elliptic curve of degree 6 can be represented as a codimension 2 linear section of $P^1 \times P^1 \times P^1$, or as a codimension 3 linear section of $P^2\times P^2$. – Sasha Mar 8 '16 at 8:33
• Certainly the closure of $X_n$ is not always smooth. This is not even true for the Hilbert scheme of genus $0$ curves, which is a big part of the reason that we use the moduli spaces of stable maps. By the way, in your description of $X_4$, what happens for a pencil of quadrics that contains a $2$-plane in its base locus? Are you certain you wrote down the Hilbert scheme, rather than one of its birational modifications (e.g., obtained by varying the Bridgeland stability condition)? – Jason Starr Mar 8 '16 at 9:31
• Just to answer my own question: it seems that the locus of pencils with a planar base locus contains the transform of several different loci in the Hilbert scheme. First, the union of a plane cubic and a line skew to the plane (intersecting the cubic) transforms to such. Next, plane quartics with two nodes and two embedded points at the nodes transform to pencils of quadrics whose base locus is a 2-plane with an embedded line. Also, plane quartics with a tacnode (or ramphoid cusp) transform to pencils of quadrics whose base locus is a union of a 2-plane and a skew line. – Jason Starr Mar 8 '16 at 10:02
• @Sasha Sure but such elliptic curves have a lot of extra structure. arxiv.org/pdf/1306.4424v1.pdf Theorem 2.1 describes this extra structure for the first case and Theorem 3.1 for the second case. – Will Sawin Mar 8 '16 at 14:05
• Even if you probably know that, I would mention the work of Tom Fisher at Cambridge dpmms.cam.ac.uk/~taf1000/research.html He studied these questions a lot, but he does not seem to have much for n>5. – Abdelmalek Abdesselam Mar 8 '16 at 15:05