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

Description of constant

For each integer $d \ge 3$, let $\ell_0(d)$ denote the maximal number of lines contained in a smooth surface of degree $d$ in $\mathbb P^3_{\mathbb C}$.

We define \(C_{76} := \limsup_{d\to\infty}\frac{\ell_0(d)}{d^2}.\)

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 \(3 \le C_{76} \le 11.\) [BS2007-lb-3d2] [BR2023-ub-11d2]

The current best general upper bound is due to Bauer and Rams, \(\ell_0(d) \le 11d^2 - 30d + 18,\) while a standard infinite lower-bound family comes from surfaces of the form $\varphi(x,y)-\psi(z,t)=0$, which give at least $3d^2$ lines for every $d$. [BR2023-ub-11d2] [BS2007-lb-3d2]

Bauer and Rams also note that the exact fixed-degree maximum remains unknown for every $d \ge 5$. [BR2023-ub-11d2]

Known upper bounds

Known lower bounds

Additional comments and links

• Classical-family ceiling. Boissière and Sarti show that the two classical constructions they analyze—surfaces $\varphi(x,y)-\psi(z,t)=0$ and cyclic $d$-covers of $\mathbb P^2$ branched over a smooth plane curve—do not exceed the $3d^2$ threshold. [BS2007-classical-cap]

• Sporadic fixed-degree improvements. They also construct a symmetric octic with $352$ lines. Thus specific degrees can substantially outperform the baseline value $3d^2$ coming from the standard infinite family. [BS2007-octic-352] [BS2007-lb-3d2]

• Fixed-degree open problems. Even in degree $5$ one currently only has the bound $\ell_0(5)\le127$, and Bauer–Rams emphasize that the exact fixed-degree maximum is still open for every $d \ge 5$. [BR2023-quintic-segre] [BR2023-ub-11d2]

• 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 $\ell(S)\le d^2(d^2-3d+3)$, with equality attained precisely in positive characteristic for certain Fermat surfaces with $d=p^e+1$. Along this infinite family, $\ell(S)/d^2=d^2-3d+3\to\infty$, so removing the characteristic-$0$ hypothesis would make the corresponding limsup infinite. [PRS2024-maximal-lines]

References

• [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
  • [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 $P^3(\mathbb C)$ of a fixed degree $d \ge 5$ remains unknown (see [18], [3], [12], [6]). In the case of smooth quintic surfaces the proof of the inequality $\ell(X_5) \le 127$ can be found in the recent paper [16], whereas (until now) the best bound for smooth complex surfaces of degree $d \ge 6$ has been the inequality $(1)\ \ell(X_d) \le (d-2)(11d-6)$ that was stated by Segre in [18, § 4].”
  • [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 $d \ge 3$ over a field of characteristic $0$ or of characteristic $p > d$. Let $\ell(X_d)$ be the number of lines that the surface $X_d$ contains. Then the following inequality holds $(2)\ \ell(X_d) \le 11d^2 - 30d + 18$. This result provides the lowest known bound on the number of lines lying on a degree-$d$ surface for $d \ge 6$. Still, the question what is the maximal number of lines on smooth projective surfaces of a fixed degree $d \ge 5$ remains open.”
  • [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: $(3)\ \ell(X_d) \le d(11d-24)$ ([4, p. 106]), who used ideas coming from Salmon ([4, p. 95], [17]).”
• [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
  • [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: • $N_d = 3d^2$ for $d \ge 3$, $d \neq 4, 6, 8, 12, 20$; • $N_4 = 64$, $N_6 = 180$, $N_8 = 256$, $N_{12} = 864$, $N_{20} = 1600$.”
  • [BS2007-classical-cap] loc: arXiv v1 PDF p.11, §4 (Proposition 4.2), file math/0606100v1.pdf
    quote: “Then $S$ contains exactly $\beta \cdot d$ lines. In particular, it contains no more than $3d^2$ lines.”
  • [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.”
• [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
  • [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 $d^2(d^2 - 3d + 3)$ lines. Furthermore, $S$ contains exactly $d^2(d^2 - 3d + 3)$ lines if and only if (i) $k$ has characteristic $p > 0$; (ii) $d = p^e + 1$ for some $e \in \mathbb N$; and (iii) $S$ is projectively equivalent to the Fermat surface defined by $(1)\ x^{p^e+1} + y^{p^e+1} + z^{p^e+1} + w^{p^e+1} = 0$.”

Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.