On the geography of log-surfaces

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.

💡 Research Summary

The paper is a comprehensive survey and research article on the geography of log‑surfaces, i.e. pairs (X,D) where X is a smooth projective surface and D is a reduced non‑empty divisor with simple normal crossings. It is divided into two main parts.

The first part deals with log‑surfaces of the form (ℙ²,C) where C is a reduced curve arrangement consisting of smooth rational components and having only ordinary singularities (ordinary k‑fold points). After recalling the basic definitions of logarithmic differentials, logarithmic Chern classes, and the log‑Chern slope
E(X,D)=c₁²(X,D)/c₂(X,D), the authors present the Miyaoka‑Sakai inequality c₁²≤3c₂ for log‑surfaces with semi‑stable boundary and Kodaira dimension two. They then introduce the notion of a transversal arrangement: all components are smooth, any two are either disjoint or meet transversally, and the union is connected. By blowing up all points of multiplicity ≥3 one obtains a log‑surface (Y, eD) with simple normal crossing boundary. Explicit formulas (2.2) and (2.3) express c₁²(Y,eD) and c₂(Y,eD) in terms of the number n of components, the genera g(C_i), and the numbers t_k of k‑fold points.

A central example is Hirzebruch’s construction using a full abelian (Kummer) cover of ℙ² branched along a line arrangement L with d≥6 lines and no pencil component (t_d=0). The map f:ℙ²→ℙ^{d‑1} given by the defining linear forms of the lines is composed with the Kummer map K_m^n of exponent n≥2. The fiber product X_n = {(x,y) | f(x)=K_m^n(y)} is a complete intersection of dimension two, singular precisely at the points of L of multiplicity ≥3. After desingularisation one obtains a smooth surface Y_n. The authors quote Hirzebruch’s formulas for K_{Y_n}² and e(Y_n) expressed through n, d, the total number f₀=∑{r≥2} t_r of multiple points, the weighted sum f₁=∑{r≥2} r t_r, and the number t₂ of double points. These calculations show that for line arrangements the log‑Chern slope never exceeds 5/2, and for more general rational curve arrangements the empirical upper bound appears to be 8/3.

The second part contains original results concerning log‑surfaces built from a complex projective K3 surface X (so c₁(X)=0, χ_{top}(X)=24) together with a rational curve arrangement D = ∪{i=1}^n C_i that is transversal. Because each C_i is a smooth rational curve on a K3 surface, its self‑intersection is C_i² = –2. Blowing up the k‑fold points (k≥3) yields a log‑surface (Y, eD) with simple normal crossing boundary. The authors compute
c₁²(Y,eD) = (∑ C_i)² = –2n + 2∑{i<j} C_i·C_j,
c₂(Y,eD) = 24 + ∑_{k≥2} (k−1) t_k,
where t_k counts the ordinary k‑fold points of the original arrangement. Consequently the log‑Chern slope is
E =