On the geography of log-surfaces

Technical Summary: On the geography of log-surfaces

Problem Statement
This survey addresses the "geography problem" for log-surfaces, a fundamental issue in the theory of complex algebraic surfaces. The core problem seeks to determine which pairs of Chern numbers $(c_1^2, c_2)$ can be realized by minimal complex surfaces of general type, and more specifically, by log-surfaces. A log-surface is defined as a pair $(X, D)$, where $X$ is a smooth projective surface and $D$ is a reduced boundary divisor with simple normal crossings. The authors focus on the "log-Chern slope," defined as $E(X, D) = c_1^2(X, D) / c_2(X, D)$.

The paper investigates the range of possible values for this slope, particularly in the context of the Miyaoka–Sakai inequality, which states that for log-surfaces with semi-stable boundaries and logarithmic Kodaira dimension 2, $c_1^2 \leq 3c_2$. The authors aim to understand the geography of log-surfaces where the Chern numbers are determined by the combinatorics of curve arrangements, specifically focusing on arrangements of smooth rational curves. A central question (Problem 2.13) is whether the log-Chern slope for arrangements of rational curves is bounded by $8/3$, and under what conditions it might reach the theoretical maximum of 3.

Methodology
The authors employ a combination of classical algebraic geometry, combinatorial topology, and computational number theory.

1. Log-Geometric Framework: The paper establishes the necessary definitions for log-Chern numbers $c_1^2(X, D)$ and $c_2(X, D)$ using the sheaf of logarithmic differentials $\Omega^1_X(\log D)$. For a transversal arrangement of curves $D$ on a surface $X$, the authors utilize a construction involving the blow-up of $X$ at intersection points of multiplicity $k \geq 3$ to form a log-surface $(Y, \tilde{D})$. Explicit formulas (Proposition 2.12) relate the log-Chern numbers of $(Y, \tilde{D})$ to the combinatorial data of the arrangement (number of curves, geometric genera, and counts of $k$-fold intersection points $t_k$).

2. Combinatorial Analysis of Arrangements:

   • Line Arrangements: The authors review the characteristic numbers of line arrangements in $\mathbb{P}^2$, drawing heavily on Hirzebruch's work on abelian covers. They analyze inequalities such as Hirzebruch's inequality and the Melchior inequality to bound the characteristic numbers $\gamma(L)$.
   • Conic-Line Arrangements: The study is extended to arrangements of lines and smooth conics. The authors utilize a Hirzebruch-type inequality for conic-line arrangements (Theorem 4.1) to derive bounds on their characteristic numbers.
   • K3 Surfaces: The second part of the paper shifts to arrangements of rational curves on smooth complex projective K3 surfaces. Here, the geometry differs significantly from $\mathbb{P}^2$ due to the rigidity of rational curves and negative self-intersections. The authors derive new combinatorial constraints (Theorem 5.2) specific to K3 surfaces.

3. Computational and Arithmetic Methods: To verify combinatorial configurations on specific K3 surfaces (e.g., Fermat and Schur quartics), the authors employ a "good reduction" method. They define the scheme of lines on a quartic surface over a number field and prove that for primes of good reduction, the incidence graph of the lines is preserved modulo $p$ (Proposition 5.12, Corollary 5.13). This allows them to compute intersection counts ($t_k$) and log-Chern slopes using finite field arithmetic (via Magma) rather than complex symbolic computation, which is often intractable.

Key Contributions and Results

• Line Arrangements in $\mathbb{P}^2$:

  • The paper clarifies the distinction between real and complex line arrangements. For real arrangements, the characteristic number is bounded by $5/2$, with equality if and only if the arrangement is simplicial (Theorem 3.5).
  • For complex arrangements, the upper bound is $8/3$. The authors provide a proof (Theorem 3.8) that equality holds if and only if the arrangement is the dual Hesse arrangement (9 lines, 12 triple points).
  • They discuss the CEVA family of arrangements, which approaches $5/2$ from above, and propose Conjecture 3.16: for any $\epsilon > 0$, there are only finitely many complex line arrangements with slope $> 2.5 + \epsilon$.

• Conic-Line Arrangements:

  • The authors prove that for arrangements of $d$ lines and $k \geq 3$ smooth conics with ordinary singularities, the characteristic number is strictly less than $8/3$ (Theorem 4.3).
  • Under the stricter condition that only double and triple points exist, the slope is strictly less than $5/2$ (Proposition 4.6).
  • They present the "Klein arrangement" of conics and lines as an example with a slope of $\approx 2.512$, currently the highest known value in this class.

• Log-Surfaces on K3 Surfaces:

  • The authors demonstrate that the intuition from $\mathbb{P}^2$ does not directly transfer to K3 surfaces. They provide explicit examples where the log-Chern slope reaches $8/3$ (Proposition 5.3) and values between $5/2$ and $8/3$ (Proposition 5.6) using configurations of lines on Fermat and Schur quartic surfaces.
  • Characterization of Slope 3: A major theoretical contribution is Theorem 5.8, which provides a necessary and sufficient combinatorial condition for a log-surface $(Y, \tilde{C})$ derived from a K3 surface to have a log-Chern slope of exactly 3. The condition requires that the arrangement has no intersection points of multiplicity $r \geq 4$ and satisfies the equation $4n = 72 + t_2 + t_3$, where $n$ is the number of curves.
  • Slope 2: The authors show that for arrangements with only double points on a K3 surface, the slope is exactly 2 if and only if the number of curves is 24 (Proposition 5.10).

• Methodological Tool: The paper formalizes the use of good reduction of line schemes to reconstruct incidence graphs of lines on quartic K3 surfaces, providing a robust computational framework for verifying combinatorial data (Section 5.2).

Significance and Claims
The authors position this work as a survey that consolidates existing knowledge on the geography of log-surfaces while introducing original results, particularly regarding K3 surfaces. They claim that the setting of log-surfaces determined by curve arrangements is "not well explored" and that their work provides a "platform for further research."

The paper highlights the difficulty of finding surfaces with log-Chern slopes close to 3, noting that in their practical experience, such configurations are "extremely difficult" to find. The significance of their results lies in:

1. Providing explicit combinatorial conditions (e.g., Theorem 5.8) that characterize extremal cases (slope 3) in the K3 setting.
2. Offering concrete counter-examples and bounds that refine the understanding of the Miyaoka–Sakai inequality in the context of log-surfaces.
3. Demonstrating that while the upper bound of $8/3$ is achievable in $\mathbb{P}^2$ (dual Hesse) and on K3 surfaces, the landscape of achievable slopes is highly sensitive to the underlying surface and the type of curve arrangement.

The authors do not claim to have solved the general geography problem for all log-surfaces but rather to have mapped out specific, combinatorially defined families and identified the boundaries of their possible Chern slopes. They conclude with Conjecture 4.10, suggesting that for conic-line arrangements, slopes significantly exceeding $2.5$ are rare, mirroring the conjecture for line arrangements.