Add asymptotic line-count constant for smooth degree-$d$ surfaces in $\mathbb{P}^3$ to README

Author: damek

README.md

@@ -106,6 +106,7 @@ We are arbitrarily numbering the constants as $C_{1}$, $C_{2}$, etc., mostly bas
 | [73](https://teorth.github.io/optimizationproblems/constants/73a.html) | Flatness constant in dimension 3 | $2+\sqrt{2}$ | $<3.972$ |
 | [74](https://teorth.github.io/optimizationproblems/constants/74a.html) | 10-point multi-point Seshadri constant on $\mathbb{P}^2$ | $\frac{117}{370}$ | $\frac{1}{\sqrt{10}}$ |
 | [75](https://teorth.github.io/optimizationproblems/constants/75a.html) | Metric TSP subtour-LP integrality-gap constant | $\frac{4}{3}$ | $\frac{3}{2} - 2.18 \cdot 10^{-34}$ |
+| [76](https://teorth.github.io/optimizationproblems/constants/76a.html) | Asymptotic line-count constant for smooth degree-$d$ surfaces in $\mathbb{P}^3$ in characteristic $0$ | 3 | 11 |
 
 
 ## Recent progress

constants/76a.md

@@ -0,0 +1,81 @@
+# 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
+$$
+3 \le C_{76} \le 11.
+$$
+<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,
+$$
+\ell_0(d) \le 11d^2 - 30d + 18,
+$$
+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 $d$. <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 $d \ge 5$. <a href="#BR2023-ub-11d2">[BR2023-ub-11d2]</a>
+
+## Known upper bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| $\ell_0(d) \le d(11d-24)$ | <a href="#BR2023">[BR2023]</a> | Historical Clebsch bound, recorded in Bauer–Rams; it already implies $C_{76} \le 11$. <a href="#BR2023-ub-clebsch">[BR2023-ub-clebsch]</a> |
+| $\ell_0(d) \le (d-2)(11d-6)$ | <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 $d \ge 6$ until 2023. <a href="#BR2023-quintic-segre">[BR2023-quintic-segre]</a> |
+| $\ell_0(d) \le 11d^2 - 30d + 18$ | <a href="#BR2023">[BR2023]</a> | Current best general upper bound in characteristic $0$, valid for $d \ge 3$. <a href="#BR2023-ub-11d2">[BR2023-ub-11d2]</a> |
+
+## Known lower bounds
+
+| Bound | Reference | Comments |
+| ----- | --------- | -------- |
+| $\ell_0(d) \ge 3d^2$ | <a href="#BS2007">[BS2007]</a> | Achieved by the classical family $\varphi(x,y)-\psi(z,t)=0$; within that family the exact count is $3d^2$ except for the finitely many exceptional degrees $4,6,8,12,20$. Therefore $C_{76} \ge 3$. <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 $\varphi(x,y)-\psi(z,t)=0$ and cyclic $d$-covers of $\mathbb P^2$ branched over a smooth plane curve—do not exceed the $3d^2$ threshold. <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 $3d^2$ 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 $\ell_0(5)\le127$, and Bauer–Rams emphasize that the exact fixed-degree maximum is still open for every $d \ge 5$. <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 $\ell(S)\le d^2(d^2-3d+3)$, 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. <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](https://doi.org/10.2422/2036-2145.202111_010). arXiv PDF: [1902.05133v2](https://arxiv.org/pdf/1902.05133v2.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Thomas+Bauer+Slawomir+Rams+Counting+lines+on+projective+surfaces)
+    - <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 $P^3(\mathbb C)$ 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].”
+    - <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 $d \ge 3$ 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.”
+    - <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: $(3)\ \ell(X_d) \le d(11d-24)$ ([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](https://doi.org/10.2422/2036-2145.2007.1.03). arXiv PDF: [math/0606100v1](https://arxiv.org/pdf/math/0606100v1.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Samuel+Boissiere+Alessandra+Sarti+Counting+lines+on+surfaces)
+    - <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: • $N_d = 3d^2$ for $d \ge 3$, $d \neq 4, 6, 8, 12, 20$; • $N_4 = 64$, $N_6 = 180$, $N_8 = 256$, $N_{12} = 864$, $N_{20} = 1600$.”
+    - <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 $\beta \cdot d$ lines. In particular, it contains no more than $3d^2$ 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](https://doi.org/10.48550/arXiv.2406.15868). arXiv PDF: [2406.15868v2](https://arxiv.org/pdf/2406.15868v2.pdf). [Google Scholar](https://scholar.google.com/scholar?q=Janet+Page+Tim+Ryan+Karen+E.+Smith+Smooth+Surfaces+with+Maximal+Lines)
+    - <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 $d^2(d^2 - 3d + 3)$ 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.