Research database

Extremal Halphen Surfaces

Explicit plane catalogue of elliptically admissible extension orbits.

26models

2–6indices

12root types

Uniform catalogue convention

Every one of the 26 entries contains the same five blocks, in the same order: field, parameters, and cubic, ordered cluster, infinitely-near data, normal class, extension, and verification, and classes of the \((-2)\)-curves.

The class matrix always has ten columns, in the ordered basis \((L,E_1,\ldots,E_9)\). Its number of rows is the number of prime \((-2)\)-components in the affine ADE configuration.

Notation. \[ D=dL+e_1E_1+\cdots+e_9E_9,\qquad K_X=-3L+E_1+\cdots+E_9. \] \[ \mathscr G(g_1,\ldots,g_7)=g_1x^2y+g_2x^2z+g_3xy^2+g_4xyz+g_5xz^2+g_6y^2z+g_7yz^2. \]

Curve, normal class, and extension notation

Throughout the catalogue, \(C\) denotes the displayed smooth plane cubic; \(E\) denotes it when given in Weierstrass form, and \(G\) denotes its defining polynomial. The normal class and the finite lattice quotient are

\[ N=\mathcal O_C(3)(-p_1-\cdots-p_9)\in\operatorname{Pic}^0(C),\qquad H=K_X^\perp/\langle D_1,\ldots,D_r\rangle. \]

Here \(D_i\) are the rows of the class matrix. Restriction to \(C\) sends \([-K_X]\in H\) to \(N\); coordinates in a Smith basis of \(H\) refer to that basis. In the index-five entries, \(h\) denotes \(N\) in additive notation. The normal class has order \(m\), the Halphen index, on the admissible open of each family.

The symbol \(\xi\) denotes an extension class in the presentation specified for that family. Orbit labels are interpreted within the relevant root type:

sp: Split extension.
ns: Nonsplit extension.
cyc: Orbit with cyclic middle group.
mid: Intermediate orbit in the index-four cyclic-quotient families.
pair / fixed: The two-element nonzero orbit / fixed nonzero orbit for \(D_6+2A_1\).
unique nonzero: The unique nonzero extension orbit for \(2D_4\).
other 2 / ord 4: The nondistinguished order-two orbit / order-four orbit for \(2A_3+2A_1\).
\(\xi=3\) / \(\xi=\pm2\): The indicated orbits for the index-six family.

Infinitely-near points and recursive charts

The relation \(p_i\prec p_j\) means that \(p_j\) lies above \(p_i\); chains run from the proper point to the successive infinitely-near centers. Subscripts label the centers and do not always give the blow-up order. Repeated projective coordinates record the same image in the plane.

Every infinitely-near center lies on the strict transform of \(C\). Translate the stated proper center to \((u_0,v_0)=(0,0)\). For each listed tangent \([\alpha_j:\beta_j]\), the next chart, centered at the next point, is

\[ \begin{cases} u_j=u_{j+1},\quad v_j=u_{j+1}(\beta_j/\alpha_j+v_{j+1}),&\alpha_j\ne0,\\ u_j=u_{j+1}v_{j+1},\quad v_j=u_{j+1},&\alpha_j=0. \end{cases} \]

These coordinate choices fix the signs and scales used for all subsequent tangents.

Verification scope and sources

The audit of 19 September 2026 checks all 26 matrices against their cited sources and independently verifies all 268 component classes, all 60 affine fibre relations, the middle groups, and the indices. Geometric certificates were also rerun; each entry states the scope of its checks.

For nine incidence models of indices 4 or 6, the rerun verifies the specified finite-field specializations. This alone is not an independent verification of every generic characteristic-zero assertion. Geometric integrality is asserted on a suitable nonempty open, which may require more than smoothness of the cubic and distinctness of the proper centers.

Read the audit, certificate list, and source notes.

Models

Root type	Index	Extension orbit	Middle group	Components	Lines