---
title: Burkhardt Quartic Threefold
url: https://www.emergentmind.com/topics/burkhardt-quartic-threefold
type: topic
---

# Burkhardt Quartic Threefold

The Burkhardt quartic threefold is a singular rational Fano hypersurface of degree four in projective four-space, distinguished by its maximal symmetry and deep connections to abelian surfaces with level-3 structures, modular forms, birational geometry, and the representation theory of finite simple groups. Defined by a unique quartic invariant under the projective symplectic group $\PSp_4(\mathbb{F}_3)$, the Burkhardt quartic serves as a fundamental object in both algebraic geometry and arithmetic geometry, acting as a fine moduli space for principally polarized abelian surfaces with full level-3 structure and as a testing ground for the theory of birational rigidity.

## 1. Defining Equations and Normal Forms

The Burkhardt quartic $B \subset \mathbb{P}^4$ is classically given by the vanishing of the quartic polynomial
\[
F(y_0, y_1, y_2, y_3, y_4) = y_0^4 + y_0(y_1^3 + y_2^3 + y_3^3 + y_4^3) + 3 y_1 y_2 y_3 y_4 = 0.
\]
This realizes $B$ as a hypersurface in $\mathbb{P}^4$, invariant under an irreducible $5$-dimensional representation of $\PSp_4(\mathbb{F}_3)$ [1705.09006].

An equivalent symmetric form arises in $\mathbb{P}^5$ with coordinates $x_0,\dots,x_5$ via
\[
\sigma_1(x) := x_0 + x_1 + x_2 + x_3 + x_4 + x_5 = 0, \qquad \sigma_4(x) := \sum_{0 \leq i<j<k<\ell \leq 5} x_i x_j x_k x_\ell = 0;
\]
thus $X = V(\sigma_1, \sigma_4) \subset \mathbb{P}^5$ defines the threefold as a quartic hypersurface cut out by intersecting with the hyperplane $\sigma_1=0$ [1606.04372, 2308.15271]. These forms are projectively equivalent to the classical Burkhardt normal form
\[
\sum_{i=0}^4 X_i^4 - 12\sum_{0 \leq i < j \leq 4} X_i^2 X_j^2 = 0.
\]
Twisted forms of the Burkhardt quartic, arising from Galois-cohomological constructions over non-algebraically closed fields, admit quartic models in $\mathbb{P}^4$ [2207.04393].

## 2. Singularities, Planes, and Topological Invariants

The Burkhardt quartic possesses exactly 45 ordinary double points (nodes), which is the maximum possible for a quartic threefold. The singular locus, which is crucial for both birational and arithmetic considerations, decomposes under the $S_6$ (or $\PSp_4(\mathbb{F}_3)$) action into two distinct orbits: one of length 30 and one of length 15 [1606.04372, 2308.15271, 2207.04393]. Each node is analytically isomorphic to the cone over a smooth quadric surface (an $A_1$ singularity).

The quartic contains exactly 40 “Jacobi planes” (or $j$-planes), each meeting precisely 9 of the 45 nodes, and this collection forms a divisor of degree $10(-K_{X_4})$. Hyperplanes containing 18 nodes are termed "Steiner hyperplanes" and split into quadruples of Jacobi planes (“tetrahedra”), with a total of 40 such configurations. The Hessian hypersurface $\operatorname{He}(B) \subset B$, a degree-10 divisor realized as the vanishing of the determinant of the $5 \times 5$ Hessian matrix, intersects $B$ in the union of the $j$-planes [1705.09006, 2207.04393].

Minimal (crepant) resolutions of $B$ yield topological invariants:
- On the smooth quartic threefold, $h^{1,1}=1$, $h^{2,1}=70$, $\mathrm{Pic}=\mathbb{Z}$.
- After resolving the 45 nodes ($\widetilde{B}$), $h^{1,1}=46$, $h^{2,1}=70$, $\mathrm{Pic}(\widetilde{B}) \simeq \mathbb{Z}^{46}$ [2207.04393].
- For the Burkhardt quartic in the Fano context, with a small resolution $\widehat{X}$, one has $h^{1,1}(\widehat{X})=16$, $h^{2,1}(\widehat{X})=0$ [1606.04372].
The Weil class group $\mathrm{Cl}(X)$ is of rank 16, generated over $\mathbb{Q}$ by classes of the 40 $j$-planes, whereas the Cartier class group $\mathrm{Pic}(X)$ is free of rank 1 [1606.04372].

## 3. Automorphism Group, Symmetry, and Representation Theory

The automorphism group of the Burkhardt quartic is the simple group $\PSp_4(\mathbb{F}_3)$, of order $25\,920$ [2207.04393, 2308.15271]. It acts transitively on the set of singular points and exhibits remarkable linear symmetry properties:
- $S_6$ acts by permutation of coordinates in the $\mathbb{P}^5$ realization.
- The group lifts to a projective representation via explicit generators constructed from discrete Fourier transforms and diagonal scale matrices [2601.05922].

Twisted Burkhardt quartics correspond to cocycle classes in $H^1(k, \PSp_4(\mathbb{F}_3))$, and twisting the irreducible $5$-dimensional representation yields new $k$-forms $V_5^\alpha$ on which the quartic invariant is unique up to scale [2207.04393]. The full automorphism group acts generically freely outside the 45 nodes [2308.15271].

## 4. Moduli Interpretations and Connections to Abelian Surfaces

A central feature of the Burkhardt quartic is its moduli-theoretic interpretation: it is birational (over $\mathbb{Z}[1/3]$) to the Satake compactification of $\mathcal{A}_2(3)$, the fine moduli space of principally polarized abelian surfaces with full level-3 structure. On the open subset $B \setminus \operatorname{He}(B)$, $B$ parametrizes Jacobians of genus-2 curves endowed with level-3 structures [2601.05922, 1705.09006].

For a point $a \in B \setminus \operatorname{He}(B)$, the corresponding fibre of the universal genus-2 curve is given explicitly as a double cover of $\mathbb{P}^1$:
\[
C_a:\quad y^2 + G_a(x)y = x^3 H_a(x),
\]
where $H_a(x)$ and $G_a(x)$ are polynomials in the coordinates of $a$ [1705.09006]. The $j$-planes correspond to cyclic order-3 subgroups of the abelian surfaces it parametrizes, and their combinatorics encode the configuration of level-3 structures and maximal isotropic subgroups under the Weil pairing.

Abelian surfaces in Hesse form admitting symmetric level-3 theta structures embed naturally into $\mathbb{P}^8$, with defining quadratic and cubic equations parameterized by points on $B$ and its dual $\widetilde{B}$ [2601.05922]. The moduli and explicit isogeny formulas for $(3,3)$-isogenies are governed by the geometry and symmetries of the Burkhardt quartic threefold.

## 5. Birational Geometry and Rigidity

The Burkhardt quartic is a prototypical example of a $\mathfrak{A}_5$-Fano threefold that is $\mathfrak{A}_5$-birationally superrigid. The key criterion is that $\mathrm{Cl}(X)^{\mathfrak{A}_5} = \mathbb{Z}[-K_X]$, precluding the existence of nontrivial $\mathfrak{A}_5$-invariant divisors. Rigorous Noether–Fano type arguments show that linearization fails for the full automorphism group except in low-rank abelian subgroups, with birational superrigidity verified through canonicity analyses of $G$-invariant mobile linear systems [1606.04372, 2308.15271].

Notably, out of 115 nontrivial conjugacy classes in $\PSp_4(\mathbb{F}_3)$, 103 are not linearizable via projective transformations, while only eight abelian subgroups are shown to be linearizable; these cases admit explicit $G$-equivariant birational maps to $\mathbb{P}^3$ [2308.15271]. The presence of K3-fixing involutions and incompressible Burnside symbols further obstructs linearizability.

## 6. Arithmetic Twists and Galois Cohomological Obstruction

Twists of the Burkhardt quartic are classified by cocycles in $H^1(k,\PSp_4(\mathbb{F}_3))$, and each admits a quartic model in $\mathbb{P}^4$ with identical singularity structure [2207.04393]. Galois-cohomological obstructions determine whether the twist $B(\alpha)$ is birational to the true moduli space of abelian varieties with level-3 structure. The obstruction $\Ob(B(\alpha))$ is the class of a Brauer–Severi variety, with order dividing $2$ and index dividing $4$ in the Brauer group $\mathrm{Br}(k)$.

If $\Ob(B(\alpha))=0$, the twist is unirational, and Zariski-dense rational points exist in the nonsingular locus; for nonzero obstruction, only Kummer-level degenerations occur, or, in the index 4 case, the minimal desingularization may lack rational points altogether.

## 7. Zeta Function and Rational Parametrization over Finite Fields

Over finite fields $\mathbb{F}_q$ with $3 \nmid q$, the zeta function of $B$ is given by
\[
Z(B/\mathbb{F}_q, T) = \frac{(1-qT)^{15}(1-\varepsilon\, qT)^{14}(1-T)(1-q^2T)^{10}(1-\varepsilon\, q^2T)^{6}(1-q^3T)}{1},
\]
where $\varepsilon = (\frac{q}{3})$ is the quadratic character mod 3 [1705.09006]. Minimal resolutions yield zeta functions with increased exponents reflecting the contribution of exceptional divisors.

An explicit rational parametrization of $B$ by $\mathbb{P}^3$ is established over fields of characteristic not 3, with concrete polynomial maps exhibiting both dominant morphisms and rational inverses [1705.09006].

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For further technical details and proofs, see [1606.04372], [1705.09006], [2207.04393], [2308.15271], and [2601.05922].

Source: https://www.emergentmind.com/topics/burkhardt-quartic-threefold