Superfigurations in Incidence Varieties

• Superfigurations are combinatorial linear spaces defined by inequalities where every point lies on at least 3 full lines and every full line contains at least 3 points.
• They provide a finite combinatorial core for classifying the birational types of incidence varieties, with realization spaces birational to projective spaces, genus‑1 curves, or K3 surfaces.
• Computational techniques such as Gröbner bases and variable elimination enable explicit affine modeling and classification of superfiguration realization spaces for up to 10 points.

Searching arXiv for the specified paper and closely related work on incidence varieties and configurations. Superfigurations are combinatorial linear spaces that isolate the essential birational complexity of incidence varieties of point configurations in the projective plane. In the setting of configurations of at most 10 points, an $n_3$-superfiguration is a simple rank-3 matroid, equivalently a linear space $(n,L)$, in which every point lies on at least 3 full lines and every full line contains at least 3 points. The central result is that arbitrary incidence-variety components can be reduced birationally to realization spaces of such superfigurations, and that for $n \le 10$ every irreducible realization-space component is birational to a projective space, a genus-1 curve, or a K3 surface (Isham et al., 13 Jul 2025).

1. Definitions and ambient geometry

Fix a ground set of points $\{1,2,\dots,n\}$. A collection $L$ of subsets, called lines, makes $(n,L)$ into a linear space if two conditions hold: any two distinct points lie on exactly one $\ell \in L$, and every $\ell \in L$ has at least two points. An $n_3$-configuration is the special case in which every point lies on exactly 3 lines and every line contains exactly 3 points. By contrast, an $n_3$-superfiguration is defined by inequalities rather than equalities: every point lies on at least 3 full lines, and every full line contains at least 3 points (Isham et al., 13 Jul 2025).

The same notion can be expressed in matroid language. An $(n,L)$0-superfiguration is a simple rank-3 matroid on $(n,L)$1 elements in which every element has rank-1 hyperplane-degree $(n,L)$2 and every rank-1 flat has size $(n,L)$3. This formulation is useful because it connects the incidence problem to combinatorial geometry and to the structure theory of rank-3 matroids.

Given any collection $(n,L)$4 of subsets of $(n,L)$5, the associated incidence variety

$(n,L)$6

is cut out by the collinearity conditions

$(n,L)$7

for each $(n,L)$8-subset $(n,L)$9. Strong realizations impose injectivity and require that no other triple be collinear; the quotient by $n \le 10$0 is the realization space $n \le 10$1. The distinction between $n \le 10$2 and $n \le 10$3 is structural: the former records point tuples satisfying prescribed determinants, whereas the latter removes projective equivalence and excludes accidental collinearities.

2. Enumeration for at most 10 points

For $n \le 10$4, there are exactly 163 isomorphism classes of $n \le 10$5-superfigurations. This enumeration comes from computer and classical methods and provides the finite combinatorial input for the birational classification (Isham et al., 13 Jul 2025).

$n \le 10$6	Number of isomorphism classes	Remarks
7	1	the Fano plane
8	1	the Möbius–Kantor configuration
9	10	includes the Pappus configuration
10	151	ten classical $n \le 10$7-configurations and 141 “extra” superfigurations

The unique $n \le 10$8 case is the Fano plane, with

$n \le 10$9

The unique $\{1,2,\dots,n\}$0 case is the Möbius–Kantor configuration, with

$\{1,2,\dots,n\}$1

Among the ten $\{1,2,\dots,n\}$2 superfigurations, the Pappus configuration is represented by

$\{1,2,\dots,n\}$3

The $\{1,2,\dots,n\}$4 case is combinatorially richer. Ten of the 151 classes are the classical $\{1,2,\dots,n\}$5-configurations, while 141 are additional superfigurations. Representative examples include a special Desargues superfiguration in which point $\{1,2,\dots,n\}$6 lies on the base line $\{1,2,\dots,n\}$7, a “modular” superfiguration related to $\{1,2,\dots,n\}$8, the “starfish” five-point-line superfiguration, and an “anti-Pappian” unrealizable superfiguration. The appearance of these extras shows that the class of superfigurations is strictly larger than the class of classical $\{1,2,\dots,n\}$9-configurations.

3. Reduction from arbitrary incidence data

The role of superfigurations is determined by a reduction theory for incidence varieties. Proposition 2.5 states that if $L$0 is any set of collinearity conditions on $L$1 points, then every irreducible component of $L$2 is isomorphic to a component of $L$3 for some linear space $L$4 with $L$5. Thus arbitrary incidence data can first be replaced by linear spaces without losing irreducible components (Isham et al., 13 Jul 2025).

The next step is Glynn’s lemma in birational form. If $L$6 is a linear space that is not an $L$7-superfiguration, meaning that some point lies on at most 2 lines or some line has at most 2 points, then every component of $L$8 or of the strong incidence variety $L$9 is birational to

$(n,L)$0

where $(n,L)$1 is a component of an incidence variety for a linear space on $(n,L)$2 points and $(n,L)$3. In effect, one can inductively peel off points of low degree.

Proposition 2.14 packages these reductions into the decisive structural statement. For any incidence-variety component $(n,L)$4 on $(n,L)$5 points, $(n,L)$6 is birational either to a projective space $(n,L)$7 with $(n,L)$8, or to

$(n,L)$9

where $\ell \in L$0 is a component of the realization space $\ell \in L$1 of some $\ell \in L$2-superfiguration $\ell \in L$3 on $\ell \in L$4 points, and $\ell \in L$5. The upshot is that the only new building blocks in the birational classification come from realization spaces of $\ell \in L$6-superfigurations. A common misconception is therefore excluded: arbitrary collections of collinearity constraints do not generate essentially new birational types beyond those already present in superfigurations.

4. Normal forms and computational analysis

The realization spaces of superfigurations are analyzed by fixing a combinatorial V-shaped frame. Once such a frame is chosen for a superfiguration $\ell \in L$7 on $\ell \in L$8 points, four of the points can be normalized by a projective transformation. The remaining points are described by affine coordinates $\ell \in L$9 in $\ell \in L$0, leading to an affine $\ell \in L$1-scheme

$\ell \in L$2

where $\ell \in L$3 is generated by the $\ell \in L$4 minors $\ell \in L$5 for each combinatorially collinear triple $\ell \in L$6 of $\ell \in L$7 (Isham et al., 13 Jul 2025).

The computational procedure is explicit. First, in Magma/Sage, one forms the ideal $\ell \in L$8 of all collinearity determinants. Second, one computes a Gröbner basis of $\ell \in L$9, then eliminates variables by successive substitutions to get a simpler isomorphic presentation $n_3$0 in fewer variables. Third, one passes to the reduced scheme

$n_3$1

Fourth, one decomposes into $n_3$2-components and computes dimensions and, for curves, geometric genus or identifies the component as $n_3$3 or genus-1.

This procedure was carried out for all 141 of the extra 10-point superfigurations and the 12 cases with $n_3$4. The latter had already been handled by Iampolskaia–Skorobogatov–Sorokin. The computational approach is notable because the classification is not merely existential: each superfiguration is treated at the level of explicit affine equations and component decompositions.

5. Birational types of realization-space components

For every $n_3$5-superfiguration $n_3$6 with $n_3$7, each irreducible $n_3$8-component of its realization space $n_3$9 is a Zariski open subset of exactly one of three types: a projective space $n_3$0 with $n_3$1, a genus-1 curve, or a K3 surface (Isham et al., 13 Jul 2025).

The small-$n_3$2 cases are uniform. For all superfigurations with $n_3$3, every component is $n_3$4 with $n_3$5. The ten classical $n_3$6-configurations are more varied: Desargues gives $n_3$7, five of them give $n_3$8, and four give open K3 surfaces. Those four K3 cases are precisely the classical $n_3$9-configurations of discriminant $(n,L)$00 studied by Sink, and the K3 surfaces have Picard rank $(n,L)$01.

The 141 additional 10-point superfigurations contribute only lower-dimensional birational types: each component of $(n,L)$02 is open in $(n,L)$03 with $(n,L)$04 or is a genus-1 curve. In the genus-1 cases, one checks over $(n,L)$05 that there is a rational point, so each such component is an elliptic curve over $(n,L)$06. Their Cremona labels are 11a3, 14a4, 15a8, and 37a1.

A further refinement concerns degenerate 10-point examples. In a few such superfigurations one sees 2-dimensional components birational to $(n,L)$07, as in the special Desargues case. Apart from the four K3 cases and Desargues’s $(n,L)$08, only $(n,L)$09, $(n,L)$10, $(n,L)$11, or genus-1 curves occur.

6. Representative structures and conceptual clarifications

Several representative line sets illustrate the range of behavior within the class. The special Desargues superfiguration has line set

$(n,L)$12

and exhibits a 2-dimensional realization component birational to $(n,L)$13 (Isham et al., 13 Jul 2025). The “modular” example related to $(n,L)$14 and the “starfish” five-point-line superfiguration show that superfigurations may include full lines with four points, so the class is not confined to the exact $(n,L)$15-by-$(n,L)$16 regularity of classical $(n,L)$17-configurations. The “anti-Pappian” unrealizable superfiguration shows, conversely, that satisfying the combinatorial superfiguration axioms does not guarantee realizability in $(n,L)$18.

Three clarifications are especially important. First, an $(n,L)$19-superfiguration is not the same as an $(n,L)$20-configuration: the former is defined by lower bounds, the latter by exact valencies. Second, incidence varieties and realization spaces are distinct moduli objects; the latter impose injectivity, exclude additional collinear triples, and quotient by $(n,L)$21. Third, the reduction theorems imply that superfigurations are not merely one family among many but the only source of genuinely new birational geometry in the classification of incidence varieties on at most 10 points.

Altogether, the classification completes the birational description of all components of all incidence varieties definable with at most 10 points. Superfigurations serve as the finite combinatorial core of that description: 163 isomorphism classes, explicit computational models for their realization spaces, and a trichotomy of birational types consisting of projective spaces, genus-1 curves, and K3 surfaces (Isham et al., 13 Jul 2025).