The 27 Questions

This page summarizes each of the 27 questions to which this wiki is dedicated. Each question has its own wiki-page with more details; click on the titles below to access these pages.

All the cubic surfaces in projective three-space $\mathbb{P}^3$ form a projective space of dimension 19, this is the projectivization of the linear space of homogeneous forms of degree three in four variables.

Question 1: PGL-orbit of a cubic surface

Cubic surfaces in projective three space form a projective space of dimension 19. The symmetry group $\mathrm{PGL}(4)$ of the projective three space acts on any given cubic. What is the degree of the orbit closure of a generic cubic? Can you determine generators for the ideal of the orbit closure of a given cubic?

Question 2: Evaluate invariants of Salmon

The ring of invariants of polynomial functions on the space of cubic forms is generated by 6 polynomials. Given a cubic, devise a method to explicitly evaluate these polynomials on that cubic. This is equivalent to computing the map from the space of all cubics to the moduli space of isomorphism classes of cubics.

Question 3: Shape of the discriminant

The discriminant of a cubic quaternary form is a degree 32 polynomial in the coefficients of the cubic. How many monomials does this polynomial have? How many vertices does its Newton polytope have, i.e., how many $D$-equivalence classes of regular triangulations are there?

Question 4: Resultant of a cubic

The discriminant of a quaternary cubic form is the resultant of its four partial derivatives. Can we express the resultant of four quaternary quadrics as the determinant of an $8\times 8$ matrix?

Question 5: Higher discriminants

Assuming the answer to the previous question is affirmative, what is the moduli interpretation of the loci obtained in the space of cubics $\mathbb{P}^{19}$ by imposing rank conditions on this $8\times 8$ matrix?

Question 6: Nansen's discriminant

E.J. Nansen (1899)1 expresses the discriminant of a cubic as the determinant of a $20 \times 20$ matrix. A cubic is singular if this matrix has rank less than twenty. What is the geometric interpretation when this matrix is of lower rank?

Question 7: Equations for the locus of k-nodal cubics

In the space $\mathbb{P}^{19}$ of all cubics, the closure of the locus of all irreducible cubic surfaces with exactly $k$-nodes forms a subvariety $N_k \subset \mathbb{P}^{19}$ of codimension $k$. The degree of $N_k$ is 32, 280, 800, 305 for $k=1,2,3,4$ respectively and $N_k$ is empty for $k>4$. Can you find explicit, low degree polynomials vanishing on $N_k$ for $k=1,2,3,4$?

Question 8: Real binodal and trinodal cubics

Can we find 17 real points in $\mathbb{P}^3$ such that there are exactly 280 real distinct binodal cubics passing through them? Can we find 16 real points such that there are exactly 800 trinodal cubics passing through them? Is it possible to choose these real points so that none of these binodal or trinodal cubics are real?

Question 9: Real Cayley

Can we find 15 real points in $\mathbb{P}^3$ that lie on exactly 305 real Cayley symmetroids?

Question 10: Tropical enumerative geometry

Can we derive the numbers 280, 800 and 305 of Question 7 tropically?

Question 11: Smooth tropical cubic surfaces from blow-up

How can we choose six points in $\mathbb{P}^2$ with integer coordinates and how can we pick a basis for the space of ternary cubic forms through these points such that the induced rational map $\mathbb{P}^2 \dashrightarrow \mathbb{P}^3$ gives rise to a smooth tropical cubic surface with respect to the 2-adic valuation? Which unimodular triangulations arise?

Question 12: Smooth tropical cubics after change of coordinates

Given a cubic surface defined over a valued field $K$, how can we decide if it defines a smooth tropical surface after a linear transformation of $\mathbb{P}_K^3$? How can we find the corresponding $4\times 4$ matrix?

Question 13: Singularities of the locus of cubics with an Eckhard point

Salmon's degree 100 invariant on quaternary cubic forms vanishes on a form if the associated surface contains three concurrant lines (i.e. an Eckhard point). The zero locus of this invariant is a degree 100 hypersurface in $\mathbb{P}^19$. What can we say about the cubic forms contained in the singular locus of this hypersurface?

Question 14: Real rank boundary

A quaternary cubic $f$ is a $4\times 4\times 4$-tensor. Generically, the complex tensor rank of $f$ is 5 but the real rank of $f$ becomes 6 if one crosses the real rank boundary studied by Michalek and Moon (2018)2. What can be said about the tropicalization of the Michalek-Moon hypersurface?

Question 15: Hessian discriminant

Anna Seigal [8] identifies the Hessian discriminant as a locus where the complex rank of $f$ jumps from $5$ to $6$. This is a hypersurface of degree $120$ in $\mathbb{P}^{19}$, invariant under $\textrm{PGL}(3)$. How to write the Hessian discriminant in terms of fundamental invariants?

Question 16: Eigenpoints of cubic surfaces

The eigenpoints of $f$ are the fixed points of the gradient map $\nabla f \colon \mathbb{P}^3 \dashrightarrow \mathbb{P}^3$. For generic cubics $f$, there are $15$ eigenpoints. They form the eigenconfiguration of the $4\times 4 \times 4$ tensor $f$. Which configurations of $15$ points in $\mathbb{P}^3$ arise as eigenpoints of a cubic surface?

Question 17: Eigendiscriminant

The eigendiscriminant is a hypersurface of degree $96$ in $\mathbb{P}^{19}$. It presents cubic surfaces that possess an eigenpoint Question 16 of multiplicity $\geq 2$. Does there exist a compact determinantal formula for the eigendiscriminant.

Question 18: Cayley cubics

Let $C$ be a smooth curve in $\mathbb{P}^3$ obtained by intersecting a quadric and a cubic. The space of cubic forms vanishing on $C$ is 5 dimensional (hence of projective dimension 4). However, generically, there are only 255 Cayley cubics containing $C$, each corresponding to the 255 double covers of the curve. There are even fewer Cayley cubics when the curve is special[5]. What explains this drop from 305 Cayley cubics we expect to find in a generic projective 4-space of cubics?

Question 19: Toric degenerations of Cox rings

This question paraphrases Problem 5.4 in [9]. It was studied in [3] but the authors left it largely unresolved. What are all the toric degenerations of Cox rings of cubic surfaces?

Question 20: Tropical basis for the universal Cox ideal

This question paraphrases Conjecture 5.3 in [6]. Can we identify a tropical basis for the universal Cox ideal of cubic surfaces?

Question 21: Tropical del Pezzo surfaces and valuations of invariants

There are two generic types of tropical del Pezzo surfaces of degree $3$, characterized by the three arrangements in [6]. Can we identfiy cubics $f$ that realize these two types by looking at the valuations of the six invariants in Question 2.

Question 22: Tropical basis for the universal Fano variety

The lines in $\mathbb{P}^3$ are points $p = (p_{12}\colon p_{13}\colon p_{14} \colon p_{23}\colon p_{24}\colon p_{34})$ in the Grassmannian $\textrm{Gr}(2,4)\subset \mathbb{P}^5$. The universal Fano variety in $\mathbb{P}^{19}\times \mathbb{P}^5$ parametrizes lines on cubic surfaces. Its ideal is generated modulo the Plücker quadric by $20$ polynomials of degree $(3,1)$ in $(p,c)$.

Can we identiy an explicit tropical basis for the universal Fano variety?

Question 23: Smooth hyperbolic cubics

A real cubic surface in $\mathbb{P}^3_{\mathbb{R}}$ has either one or two connected components. In the latter case, the cubic is hyperbolic. It bounds a convex body that is of interest in optimization.

Can we find a semialgebraic description for the set of smooth hyperbolic cubics in $\mathbb{P}^{19}_{\mathbb{R}}$? How to express this case distinction in terms of the six fundamental invariants?

Question 24: Smooth tropical cubics from Sylvestar's Pentahedral Theorem

Every cubic $f$ whose Hessian discriminant (Question 15) is nonzero has a unique representation as a sum of five third powers of linear forms, $f = l_1^3+l_2^3+l_3^3+l_4^3+l_5^3$. This is Sylvestar's Pentahedral Theorem. George Salmon [7] uses this to write the invariants.

Can we find explicit linear forms $l_i \in \mathbb{Z}[x,y,z,w]$ such that $2$-adic tropicalization of $V(f)$ is tropically smooth. Which unimodular triangulations of $3\nabla_3$ arise?

Question 25: Cubic surfaces and tropical bitangents

If we project a cubic surface $V(f)$ from a general point $p$ on that surface then we get a double-covering of $\mathbb{P}^2$ branched along a quartic curve. The $28$ bitangents of that curve are the images of the $27$ lines on $V(f)$ plus one more line which is the exceptional divisor over $p$.

Can this correspondence from $27$ to $28$ be understand in tropical geometry? In this particular, can we see the seven $4$-tuples of tropical bitangents already $\textrm{Trop}(V(f))$?

The seven $4$-tuples of bitangents are explained in [4].

Question 26: Brundu-Logar normal form

Michela Brundu and Alessandro Logar [1] offer a computational study of cubic surfaces via the normal form

$f = a_1(2x^2y−2xy^2 +xz^2 −xzw−yw^2 +yzw) + a_2(x−w)(xz+yw)+ a_3(z+w)(yw−xz) + a_4(y−z)(xz+yw) + a_5(x−y)(yw−xz).$

This amounts to fixing an $L$-set, which is a special configuration of five lines on $V(f)$.

How to compute the Brundu-Logar form in practise? Can we write $a_1,a_2,a_3,a_4,a_5$ as rational functions in $c_1,c_2,a_3,\dots,c_{20}$? What does this tell us tropically?

Question 27: Steiner representations

Here is another normal form, found in [2]. From the classical construction of Steiner sets, one shows that a general cubic surface has $120$ distinct representations
$f = l_1l_2l_3+m_1m_2m_3$, where the $l_i$ and $m_i$ are linear forms. We call this a Steiner representation of the cubic $f$.

How to compute Steiner representations in practise? Can it be tropicalized?

Bibliography
1. M.Brundu and A. Logar, Classification Of Cubic Surfaces with Computational Methods, Transformation Groups (1998), article.
2. A. Buckley and T. Košir, Determinantal Representations of Smooth Cubic Surfaces, Geometriae Dedicata (2007), article.
3. M.B. Guillén, D. Corey, M. Donten-Bury, N. Fujita and G. Merz, Khovanskii Bases of Cox–Nagata Rings and Tropical Geometry, Fields Institute Communications (2017), article.
4. Lee, H. and Len, Y., Bitangents of Non-smooth Tropical Quartics, Portugaliae Mathematica (2018), article.
5. N. Bruin and E.C. Sertöz, Prym varieties of genus four curves (2018), article.
6. Q. Ren, K. Shaw and B. Sturmfels, Tropicalization of del Pezzo surfaces, Advances in Mathematics (2016), article.
7. G. Salmon, On Quaternary Cubics, Philosophical Transactions of the Royal Society of London (1860), article.
8. A. Seigal, Ranks and Symmetric Ranks of Cubic Surfaces (2018), article.
9. B. Sturmfels and Z. Xu, Sagbi bases of Cox–Nagata rings, Journal of the European Mathematical Society (2010), article.
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