On the geography of log-surfaces

TL;DR

The paper surveys the geography problem for log-surfaces, focusing on complex log-surfaces built from line and conic-line arrangements in the plane and on K3 surfaces with rational-curve arrangements. It develops and uses log-Chern slope as a key invariant, articulates Hirzebruch-type inequalities for combinatorial configurations, and presents concrete high-slope examples including line and conic-line configurations and K3 pairs achieving slopes near or at 8/3. A central theme is understanding how combinatorics of curve arrangements controls the log-surfaces’ Chern data, with an emphasis on obtaining slopes close to the Miyaoka-Sakai bound via explicit constructions and a good-reduction framework. The work provides both theoretical criteria and computational methods (via incidence graphs and reductions) to explore and realize high-slope log-surfaces, supported by extensive explicit examples and tables.

Abstract

This survey focuses on the geometric problem of log-surfaces, which are pairs consisting of a smooth projective surface and a reduced non-empty boundary divisor. In the first part, we focus on the geography problem for complex log-surfaces associated with pairs of the form $(\mathbb{P}^{2}, C)$, where $C$ is an arrangement of smooth plane curves admitting ordinary singularities. Specifically, we focus on the case in which $C$ is an arrangement consisting of smooth rational curves as its irreducible components. In the second part, containing original new results, we study log-surfaces constructed as pairs consisting of a complex projective $K3$ surface and a rational curve arrangement. In particular, we provide some combinatorial conditions for such pairs to have the log-Chern slope equal to $3$. Our survey is illustrated with many explicit examples of log-surfaces.

Figures (3)

• Figure 1: A transversal arrangement $\mathcal{D}=\{C_1,\dots,C_n\}$ of smooth curves on a smooth surface $X$: any two distinct components are either disjoint or meet transversally, several curves may meet at one point (ordinary $k$-fold points with $k\ge2$), and the union $\bigcup_i C_i$ is connected. Blowing up the points with $k\ge3$ yields the associated log surface $(Y,\widetilde{D})$ with $\widetilde{D}$ a simple normal crossings divisor (cf. Definition \ref{['def:trans']} and Proposition \ref{['chernn']}).
• Figure 2: The polyhedral arrangement $\mathcal{P}_3$: 6 lines given by the sides of a regular triangle and its three lines of symmetry (medians). It is simplicial; the only singularities are 3 double points (the side midpoints) and 4 triple points (the three vertices and the center), so $(d,t_2,t_3)=(6,3,4)$ and $\gamma(P_3)=5/2$ (the equality case for real arrangements in Theorem \ref{['charR']}).
• Figure 3: The dual Hesse arrangement $\mathcal{L}$ of nine lines in $\mathbb{P}^2$ (incidence type $(9_4,12_3)$): all $12$ intersection points are triple (no double points), each line contains four triple points, and $\mathcal{L}$ is unique up to projective equivalence; it realizes the equality case $\gamma(\mathcal{L})=8/3$ in Theorem \ref{['somm']}.

Theorems & Definitions (70)

• Theorem 1.2
• Theorem 1.3: Persson
• Theorem 1.4
• Remark 1.5
• Definition 2.1
• Definition 2.2
• Lemma 2.3: EV
• Definition 2.4
• Definition 2.5
• Theorem 2.6: MM02