A literature search arising from the Penrose 8-conic theorem project. Answer generated by Claude Fable-5 model on August 26, 2026.

The question #

The Penrose 8-conic theorem is formulated in terms of complete conics: pairs $(Q, Q^*)$ of symmetric $3\times 3$ matrices — a point-wise conic $Q$ and a line-wise conic $Q^*$ — that are compatible in the sense

When $\lambda \neq 0$ both matrices have rank 3 and each determines the other (up to scale, $Q^* \sim \mathrm{adj}\,Q$). When $\lambda = 0$ the complete conic is degenerate. Imposing in addition the mutual adjugate-compatibility conditions

(with the zero matrix admitted), the possible rank pairs $(\mathrm{rank}\,Q, \mathrm{rank}\,Q^*)$ are exactly

The types $(1,0)$ and $(0,1)$ are a pointless line — a double line $m^2$ carrying no dual (point-wise) structure at all — and dually a lineless point — a double point $C^2$ carrying no line-wise structure.

Are complete conics of type $(1,0)$ and $(0,1)$ known in the literature? Is there a citable reference for them?

Short answer #

No. A search of the algebraic-geometry, projective-geometry, and Cayley–Klein literature turned up no treatment of the $(1,0)$ / $(0,1)$ types as complete conics. Every standard framework rules them out by definition rather than studying them. The citation situation is therefore: one can cite the literature for their exclusion, but apparently not for their existence. The names “pointless line” and “lineless point” also appear to be new. Details by tradition follow.

1. The enumerative / algebraic geometry tradition #

The classical variety of complete conics is the closure of the graph of the adjugate map $Q \mapsto \mathrm{adj}\,Q$ inside $\mathbb{P}^5 \times \mathbb{P}^{5*}$ — equivalently, the blow-up of $\mathbb{P}^5$ along the Veronese surface of double lines. Standard references:

• J. G. Semple, On Complete Quadrics I, II, J. London Math. Soc. 23 (1948), 258–267 and 27 (1952), 280–287; also Semple, Complete conics of S₂ and their model variety, Phil. Trans. Royal Soc. A (1982).
• J. A. Tyrrell, Complete quadrics and collineations in $S_n$, Mathematika 3 (1956), 69–79.
• I. Vainsencher, Schubert calculus for complete quadrics, in Enumerative Geometry and Classical Algebraic Geometry (1982).
• C. De Concini, C. Procesi, Complete symmetric varieties, Springer Lecture Notes in Math. 996 (1983) — the modern “wonderful compactification” of $SL_3/SO_3$.

In all of these, both members of the pair live in projective spaces, so neither can be zero by construction, and the orbit/boundary stratification is exactly $(3,3)$, $(2,1)$, $(1,2)$, $(1,1)$ — two boundary divisors whose intersection is the $(1,1)$ flag stratum.

There is also a geometric reason the $(1,0)$ type never forces itself on these authors: it is not a limit of regular complete conics. If regular pairs $(Q_t, Q_t^*)$ have $Q_t \to m^2$, the duals $Q_t^*$ — projectively rescaled — always accumulate on some nonzero point-pair or double point on $m$. The pointless line only exists down in the affine cone $S^2V^* \times S^2V$, where the zero matrix is admitted; projectivization erases it.

Notably, the $(1,0)$ objects are precisely the centers of the blow-up: a point of the Veronese surface in $\mathbb{P}^5$ is exactly “a double line with no dual data,” and the complete-conics construction exists to replace each such point by the $\mathbb{P}^2$ of complete conics $(m^2, B)$ lying over it. So the classical literature knows the $(1,0)$ object intimately — as the defect to be resolved, never as a citizen of the space.

2. The projective / dynamic geometry tradition #

J. Richter-Gebert, Perspectives on Projective Geometry (Springer, 2011), is the treatment closest to this project’s CompleteConic.js. His Definition 9.5 (§9.6) reads: a primal/dual pair of conics is a pair $(A, B)$ of real symmetric nonzero $3\times 3$ matrices with $AB = \lambda E$. The nonzero requirement is explicit and deliberate.

Similarly, I. Izmestiev, Spherical and hyperbolic conics (survey, 2017), §2, states flatly that “the duality between non-degenerate point conics and line conics does not extend to degenerate ones; the best one can do is to extend the relation $H_S \cdot H_{S^*} = \mathrm{Id}$ projectively” — again with both sides nonzero.

3. The Cayley–Klein tradition #

The classification of planar Cayley–Klein geometries by signature pairs (Richter-Gebert ch. 20.6; Yaglom; Struve–Struve) runs through all seven degenerate absolutes — Euclidean $(1,2)$, dual-Euclidean $(2,1)$, Galilean $(1,1)$, Minkowski, etc. — and every one has both members present. This is forced: an absolute of type $(1,0)$ would define a geometry whose angle measure is not merely degenerate but entirely undetermined, so no metric geometry ever needed the object.

4. Where the types (1,0) and (0,1) do live #

The rank-pair list above — which correctly admits $(1,0)$ but excludes $(2,0)$, since $\mathrm{adj}\,Q \sim Q^*$ with $Q^* = 0$ forces $\mathrm{rank}\,Q \leq 1$ — describes the affine multicone over the variety of complete conics. That cone appears implicitly in De Concini–Procesi’s coordinate-ring computations and in the recent Cox-ring literature (A. Massarenti, On the birational geometry of spaces of complete forms I: collineations and quadrics, Proc. London Math. Soc. 2020), but no source singles out its one-sided points as geometric objects, let alone names or classifies them.

Classically, the one-sided objects are simply: a repeated line regarded purely as a locus (a point of the rank-1 Veronese locus in $\mathbb{P}^5$, whose tangential equation / adjugate vanishes identically), and dually a repeated point regarded purely as an envelope. Classical texts (e.g. Salmon, A Treatise on Conic Sections) note the vanishing of the adjugate but do not treat the resulting “one-sided” pair as an object in its own right.

5. Caveats and conclusion #

A search cannot prove a negative. Two sources that could not be checked directly are E. Study’s early papers on systems of conics and M. Eide, The conic space (Ariadne Yearbook, 1996) — Eide, a coauthor of the 8-conic paper, would be the natural person to ask whether he handled the one-sided types there.

As far as the accessible literature goes: using the types $(1,0)$ and $(0,1)$ — and the names pointless line and lineless point — in the Penrose work would be introducing them. The citable justification is exactly the exclusions above: Semple / Tyrrell / Vainsencher / De Concini–Procesi for the projective setting, and Richter-Gebert Def. 9.5 for the explicit “nonzero” convention that the definition relaxes.

Follow-up: more information on Study’s early papers #

The question #

Section 5 above notes that E. Study’s early papers on systems of conics could not be checked directly. What more is known about Study’s early work on conics, and does it bear on the $(1,0)$ / $(0,1)$ question?

The answer #

Study turns out to be a central figure in exactly this story — his Habilitation is arguably where the “complete conic” as a pair was first made precise.

The paper itself #

• E. Study, Ueber die Geometrie der Kegelschnitte, insbesondere deren Charakteristikenproblem. This existed in two forms: his Habilitationsschrift (Leipzig, Teubner, 1885) and the journal version in Mathematische Annalen 27 (1886), pp. 58–101. Both are digitized (Deutsche Digitale Bibliothek / HathiTrust for the 1885 booklet; the Annalen volume via mathdoc/GDZ).

The context and content #

The backstory is the great enumerative-geometry controversy. Chasles (1864) had introduced the characteristics $(\mu, \nu)$ of a one-parameter system of conics and the formula $a\mu + b\nu$ for the number of conics satisfying an extra condition. Halphen, from 1876 on, produced counterexamples, which he traced to a subtlety Chasles had missed: there are not two but three kinds of degenerate conics in a system — in the rank language of this document, precisely $(2,1)$, $(1,2)$, and $(1,1)$ — and naive counting fails when a system contains the “deeper” degenerations.

In 1884, Klein (then in Leipzig) assigned the 22-year-old Study the problem of assessing the validity of Chasles’s formula. Study’s answer, in the 1885/86 paper, was conceptual rather than computational: Halphen’s counterexamples refute only one interpretation of the word “conic”. If one takes the object being counted to be the point-conic together with its envelope, on equal footing — so that degenerating families retain both the limiting locus and the limiting envelope — then Chasles’s formula is correct as stated. This is, in substance, the birth of the complete conic; Zeuthen had earlier gestured at the same idea (“regarding in an equal manner punctual and tangential properties,” which he attributed to Chasles’s own implicit viewpoint), but Study made it a precise definitional program. Study and Halphen met in Paris in 1886 (Study traveled there with Hilbert) and corresponded, but never agreed — Study insisted both interpretations were legitimate conventions; Halphen “persisted in finding nothing new or useful” in Study’s reading.

Study’s line was vindicated half a century later: van der Waerden, “Zur algebraischen Geometrie XV,” Math. Annalen 115 (1938), 645–655, gave the first rigorous proof of Chasles’s formula, working on what is essentially the complete-conics variety, and the Semple–Tyrrell–Vainsencher–De Concini–Procesi lineage descends from there. (Study also later attacked Schubert’s “principle of conservation of number” directly: Über das Prinzip der Erhaltung der Anzahl, ICM Heidelberg 1904.)

What this means for the (1,0) / (0,1) question #

Study’s paper strengthens the exclusion story rather than weakening it. His entire innovation was to define the conic-object so that the dual datum never gets lost in the limit — the three degenerate types he needed are exactly the both-sided ones $(2,1)$, $(1,2)$, $(1,1)$ that Halphen had identified. A one-sided object like $(m^2, 0)$ is precisely the “information-losing” limit his definition was engineered to prevent, and it plays no role in the counting problems the paper addresses. Caveat: this description rests on secondary accounts, not a reading of the German original, so a direct check of his degenerate-type list would require the primary source.

Where to read more #

• S. Kleiman, “Chasles’s Enumerative Theory of Conics: A Historical Introduction,” in Studies in Algebraic Geometry (MAA, 1980), pp. 117–138 — the standard mathematical account of the whole Chasles–Halphen–Study affair; the best single reference to check Study’s degenerate types against the rank table above.
• E. Casas-Alvero, S. Xambó-Descamps, The Enumerative Theory of Conics after Halphen, Springer Lecture Notes in Math. 1196 (1986) — reworks Halphen’s alternative theory rigorously and discusses the Study-vs-Halphen “conventions” issue in its introduction.
• N. Michel, “Mathematical Selves and the Shaping of Mathematical Modernism”, Isis 112 (2021) — a detailed history of the dispute, with a section specifically on Study’s epistemological position.
• Y. Hartwich, Eduard Study (1862–1930): Ein mathematischer Mephistopheles im geometrischen Gärtchen, Ph.D. dissertation, Mainz 2005 — the fullest biography of Study, covering this early work.