Orbit closure of a general cubic surface

# Setup and question

Let $V_3 \subset \mathbb{C}[x,y,z,w]$ be the space of general homogeneous forms of degree 3. For any $f \in V_3$ the group $\mathrm{PGL}(\mathbb{C},4)$ acts on $f$ by precomposition. The closure of this orbit $\overline{ \mathrm{PGL}(\mathbb{C},4) \cdot f }$ is a projective variety in $\mathbb{P}(V)$.

What is the degree of this orbit closure? Can you write down generators for the ideal of this subvariety in terms of the coefficients of $f$?

# 96120

In the paper 96120: The degree of the linear orbit of a cubic surface (https://arxiv.org/abs/1909.06620), Laura Brustenga i Moncusi, Sascha Timme, and Madeleine Weinstein find the degree to be 96120 using numerical methods.

# Possible answer for the degree

The degree of the orbit closure of the Cayley cubic to be 305, see Vainsencher "Hypersurfaces with up to six double points". The Cayley cubic is special in that it has 24 extra symmetries as opposed to the general cubic. Each of these 24 symmetries can be realized by a linear transformation, i.e., an element of $\mathrm{PGL}(\mathbb{C},4)$. It seems feasible to imagine that as a general cubic specializes to the Cayley cubic, the 24 leaves of its orbit collapses to the orbit of the Cayley cubic. Therefore, a general complementary dimensional linear space should intersect the orbit of a general cubic surface 24 times as often as it does the orbit of the Cayley cubic. The degree of the generic orbit would then be $305 \cdot 24=7320$.

# There are other full dimensional limits other than the Cayley Cubic such as the orbit of an A3 singularity (so 7320 would be strictly smaller than the correct answer)

(Hunter Spink, contact me I am working on related problems and would love to collaborate!)

Start with the Cayley Nodal Cubic $xyz+xyw+xzw+yzw=0$, and consider the deformation (where we will send t->0)

$xyz+xyw+xzw+yzw+t^2(ax^2+bz^2)(x+z)$ with $a,b$ general complex numbers (and we can add a t^3 terms to ensure that it is a general orbit limit).

Now apply the GL-change of coordinates $(x,y,z,w)->(-x,ty,-z,-ty+t^2w)$ to get $t^2(xzw+(x+w)(y^2-ax^2-bz^2))+O(t^3)$.

Clearing out the t^2 since we are working with projective coordinates and letting t->0, we get the A3 singularity from https://singsurf.org/parade/Cubics.php

# Active problem

This problem is currently being worked on by Elisa Cazzador and Bjørn Skauli, University of Oslo.
(Feel free to contact them if you would like to discuss the question!)

The main idea is to resolve the indeterminacies of the rational map $\mathbb{P}^{15} \to \mathbb{P}^{19}$ arising from the action of $\mathrm{PGL}(4)$ when we fix a general cubic form.
Once we blow-up $\mathbb{P}^{15}$ along the components of base locus, it is possible to get the related intersection degrees, thanks to a result in .
This is essentially the approach that Aluffi and Faber used in  to solve the same problem for smooth plane curves of any degree.

Bibliography
1. Aluffi, Paolo. "The enumerative geometry of plane cubics." Mathematische Annalen 289.1 (1991): 543-572.
2. Aluffi, Paolo, and Carel Faber. "Linear orbits of smooth plane curves." arXiv preprint alg-geom/9206001 (1992).