Burkhardt Quartic Threefold

• The Burkhardt quartic threefold is a singular rational Fano hypersurface defined by a unique quartic invariant with 45 nodes and maximal symmetry under PSp4(𝔽₃).
• It serves as a moduli space for principally polarized abelian surfaces with level-3 structure, linking algebraic and arithmetic geometry with birational rigidity.
• Its rich geometry features explicit symmetric forms, detailed singularity analysis, and deep connections to modular forms and finite group representation theory.

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 $\PSp_4(\mathbb{F}_3)$ (Bruin et al., 2017).

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 $\sigma_1=0$ (Cheltsov et al., 2016, Cheltsov et al., 2023). 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 $\mathbb{P}^4$ (Bruin et al., 2022).

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 (Cheltsov et al., 2016, Cheltsov et al., 2023, Bruin et al., 2022). 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 (Bruin et al., 2017, Bruin et al., 2022).

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 ($\widetilde{B}$), $h^{1,1}=46$, $h^{2,1}=70$, $\mathrm{Pic}(\widetilde{B}) \simeq \mathbb{Z}^{46}$ (Bruin et al., 2022).
• For the Burkhardt quartic in the Fano context, with a small resolution $\widehat{X}$, one has $h^{1,1}(\widehat{X})=16$, $h^{2,1}(\widehat{X})=0$ (Cheltsov et al., 2016). 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 (Cheltsov et al., 2016).

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 $25\,920$ (Bruin et al., 2022, Cheltsov et al., 2023). 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 (Decru et al., 9 Jan 2026).

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 (Bruin et al., 2022). The full automorphism group acts generically freely outside the 45 nodes (Cheltsov et al., 2023).

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 (Decru et al., 9 Jan 2026, Bruin et al., 2017).

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 $a$ (Bruin et al., 2017). 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 $\widetilde{B}$ (Decru et al., 9 Jan 2026). 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 (Cheltsov et al., 2016, Cheltsov et al., 2023).

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 $\mathbb{P}^3$ (Cheltsov et al., 2023). 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 (Bruin et al., 2022). 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 (Bruin et al., 2017). 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 (Bruin et al., 2017).

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For further technical details and proofs, see (Cheltsov et al., 2016, Bruin et al., 2017, Bruin et al., 2022, Cheltsov et al., 2023), and (Decru et al., 9 Jan 2026).