Asymptotic line-count constant for smooth degree-d surfaces in ℙ^3 in characteristic 0

Description of constant

For each integer $d \geq 3$ , let $ℓ_{0} (d)$ denote the maximal number of lines contained in a smooth surface of degree $d$ in $P_{C}^{3}$ .

We define $C_{76} := d \to \infty lim sup \frac{ℓ _{0} ( d )}{d ^{2}} .$

The constant $C_{76}$ measures the quadratic growth rate of the maximal line count on smooth complex degree- $d$ surfaces. The problem is open; the best established range is $3 \leq C_{76} \leq 11.$ <a href="#BS2007-lb-3d2">[BS2007-lb-3d2]</a> <a href="#BR2023-ub-11d2">[BR2023-ub-11d2]</a>

The current best general upper bound is due to Bauer and Rams, $ℓ_{0} (d) \leq 11 d^{2} - 30 d + 18,$ while a standard infinite lower-bound family comes from surfaces of the form $φ (x, y) - ψ (z, t) = 0$ , which give at least $3 d^{2}$ lines for every $d$ . <a href="#BR2023-ub-11d2">[BR2023-ub-11d2]</a> <a href="#BS2007-lb-3d2">[BS2007-lb-3d2]</a>

Bauer and Rams also note that the exact fixed-degree maximum remains unknown for every $d \geq 5$ . <a href="#BR2023-ub-11d2">[BR2023-ub-11d2]</a>

Known upper bounds

Bound	Reference	Comments
$ℓ_{0} (d) \leq d (11 d - 24)$	<a href="#BR2023">[BR2023]</a>	Historical Clebsch bound, recorded in Bauer–Rams; it already implies $C_{76} \leq 11$ . <a href="#BR2023-ub-clebsch">[BR2023-ub-clebsch]</a>
$ℓ_{0} (d) \leq (d - 2) (11 d - 6)$	<a href="#BR2023">[BR2023]</a>	Segre’s classical improvement; as summarized by Bauer–Rams, this was the best general bound for smooth complex degree- $d$ surfaces with $d \geq 6$ until 2023. <a href="#BR2023-quintic-segre">[BR2023-quintic-segre]</a>
$ℓ_{0} (d) \leq 11 d^{2} - 30 d + 18$	<a href="#BR2023">[BR2023]</a>	Current best general upper bound in characteristic $0$ , valid for $d \geq 3$ . <a href="#BR2023-ub-11d2">[BR2023-ub-11d2]</a>

Known lower bounds

Bound	Reference	Comments
$ℓ_{0} (d) \geq 3 d^{2}$	<a href="#BS2007">[BS2007]</a>	Achieved by the classical family $φ (x, y) - ψ (z, t) = 0$ ; within that family the exact count is $3 d^{2}$ except for the finitely many exceptional degrees $4, 6, 8, 12, 20$ . Therefore $C_{76} \geq 3$ . <a href="#BS2007-lb-3d2">[BS2007-lb-3d2]</a>

Additional comments and links

Classical-family ceiling. Boissière and Sarti show that the two classical constructions they analyze—surfaces $φ (x, y) - ψ (z, t) = 0$ and cyclic $d$ -covers of $P^{2}$ branched over a smooth plane curve—do not exceed the $3 d^{2}$ threshold. <a href="#BS2007-classical-cap">[BS2007-classical-cap]</a>

Sporadic fixed-degree improvements. They also construct a symmetric octic with $352$ lines. Thus specific degrees can substantially outperform the baseline value $3 d^{2}$ coming from the standard infinite family. <a href="#BS2007-octic-352">[BS2007-octic-352]</a> <a href="#BS2007-lb-3d2">[BS2007-lb-3d2]</a>

Fixed-degree open problems. Even in degree $5$ one currently only has the bound $ℓ_{0} (5) \leq 127$ , and Bauer–Rams emphasize that the exact fixed-degree maximum is still open for every $d \geq 5$ . <a href="#BR2023-quintic-segre">[BR2023-quintic-segre]</a> <a href="#BR2023-ub-11d2">[BR2023-ub-11d2]</a>

Why characteristic $0$ is built into the definition. Page, Ryan, and Smith prove that for smooth degree- $d$ surfaces with $d > 3$ over algebraically closed fields, one has $ℓ (S) \leq d^{2} (d^{2} - 3 d + 3)$ , with equality attained precisely in positive characteristic for certain Fermat surfaces with $d = p^{e} + 1$ . Along this infinite family, $ℓ (S) / d^{2} = d^{2} - 3 d + 3 \to \infty$ , so removing the characteristic- $0$ hypothesis would make the corresponding limsup infinite. <a href="#PRS2024-maximal-lines">[PRS2024-maximal-lines]</a>

References

<a id="BR2023"></a>[BR2023] Bauer, Thomas; Rams, Sławomir. Counting lines on projective surfaces. Annali Scuola Normale Superiore di Pisa - Classe di Scienze (5) 24 (2023), no. 3, 1285–1299. DOI: 10.2422/2036-2145.202111_010. arXiv PDF: 1902.05133v2. Google Scholar
<a id="BR2023-quintic-segre"></a>[BR2023-quintic-segre] loc: arXiv v2 PDF p.1, §1 Introduction, file 1902.05133v2.pdf quote: “By contrast, the maximal number of lines on smooth hypersurfaces in $P^{3} (C)$ of a fixed degree $d \geq 5$ remains unknown (see [18], [3], [12], [6]). In the case of smooth quintic surfaces the proof of the inequality $ℓ (X_{5}) \leq 127$ can be found in the recent paper [16], whereas (until now) the best bound for smooth complex surfaces of degree $d \geq 6$ has been the inequality $(1) ℓ (X_{d}) \leq (d - 2) (11 d - 6)$ that was stated by Segre in [18, § 4].”
<a id="BR2023-ub-11d2"></a>[BR2023-ub-11d2] loc: arXiv v2 PDF p.2, §1 Introduction (Theorem 1.1), file 1902.05133v2.pdf quote: “Theorem 1.1. Let $X_{d} \subset P^{3} (K)$ be a smooth surface of degree $d \geq 3$ over a field of characteristic $0$ or of characteristic $p > d$ . Let $ℓ (X_{d})$ be the number of lines that the surface $X_{d}$ contains. Then the following inequality holds $(2) ℓ (X_{d}) \leq 11 d^{2} - 30 d + 18$ . This result provides the lowest known bound on the number of lines lying on a degree- $d$ surface for $d \geq 6$ . Still, the question what is the maximal number of lines on smooth projective surfaces of a fixed degree $d \geq 5$ remains open.”
<a id="BR2023-ub-clebsch"></a>[BR2023-ub-clebsch] loc: arXiv v2 PDF p.2, §1 Introduction, file 1902.05133v2.pdf quote: “The first bound on the number of lines on a smooth degree- $d$ surface was stated by Clebsch: $(3) ℓ (X_{d}) \leq d (11 d - 24)$ ([4, p. 106]), who used ideas coming from Salmon ([4, p. 95], [17]).”

<a id="BS2007"></a>[BS2007] Boissière, Samuel; Sarti, Alessandra. Counting lines on surfaces. Annali Scuola Normale Superiore di Pisa - Classe di Scienze (5) 6 (2007), no. 1, 39–52. DOI: 10.2422/2036-2145.2007.1.03. arXiv PDF: math/0606100v1. Google Scholar
<a id="BS2007-lb-3d2"></a>[BS2007-lb-3d2] loc: arXiv v1 PDF p.2, §1 Introduction (statement of Proposition 3.3), file math/0606100v1.pdf quote: “Proposition 3.3 The maximal numbers of lines on $F = 0$ are: • $N_{d} = 3 d^{2}$ for $d \geq 3$ , $d \neq = 4, 6, 8, 12, 20$ ; • $N_{4} = 64$ , $N_{6} = 180$ , $N_{8} = 256$ , $N_{12} = 864$ , $N_{20} = 1600$ .”
<a id="BS2007-classical-cap"></a>[BS2007-classical-cap] loc: arXiv v1 PDF p.11, §4 (Proposition 4.2), file math/0606100v1.pdf quote: “Then $S$ contains exactly $β \cdot d$ lines. In particular, it contains no more than $3 d^{2}$ lines.”
<a id="BS2007-octic-352"></a>[BS2007-octic-352] loc: arXiv v1 PDF p.1, Abstract, file math/0606100v1.pdf quote: “We obtain in particular a symmetric octic with 352 lines.”

<a id="PRS2024"></a>[PRS2024] Page, Janet; Ryan, Tim; Smith, Karen E. Smooth Surfaces with Maximal Lines. Preprint (2024). DOI: 10.48550/arXiv.2406.15868. arXiv PDF: 2406.15868v2. Google Scholar
<a id="PRS2024-maximal-lines"></a>[PRS2024-maximal-lines] loc: arXiv v2 PDF p.1, Abstract and Theorem 1.1, file 2406.15868v2.pdf quote: “Theorem 1.1. Let $S \subseteq P^{3}$ be a smooth algebraic surface of degree $d > 3$ over an algebraically closed field $k$ . Then $S$ contains at most $d^{2} (d^{2} - 3 d + 3)$ lines. Furthermore, $S$ contains exactly $d^{2} (d^{2} - 3 d + 3)$ lines if and only if (i) $k$ has characteristic $p > 0$ ; (ii) $d = p^{e} + 1$ for some $e \in N$ ; and (iii) $S$ is projectively equivalent to the Fermat surface defined by $(1) x^{p^{e} + 1} + y^{p^{e} + 1} + z^{p^{e} + 1} + w^{p^{e} + 1} = 0$ .”

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.

Current state

The problem