Classification of low degree del Pezzo orbifolds

In this paper we classify low degree del Pezzo orbifolds with irreducible boundaries. In order to achieve desired boundaries, we classify low degree curves on low degree del Pezzo surfaces. The notion of Campana orbifolds was introduced by Campana in 2004. A del Pezzo orbifold is a Campana orbifold whose underlying surface is a del Pezzo surface. The classification is elementary applications of adjunction formula, Riemann-Roch theorem, Hodge Index theorem and Kawamata-Viehweg vanishing theorem.

💡 Research Summary

The paper undertakes a systematic classification of low‑degree del Pezzo orbifolds with an irreducible boundary divisor. A del Pezzo orbifold is defined as a Campana orbifold (X, Δ_ε) where X is a smooth del Pezzo surface and the log divisor −(K_X + Δ_ε) is ample. The authors restrict to the case where the boundary consists of a single component Δ_ε = ε D, with ε = 1 − 1/m (m ≥ 2) and D an irreducible curve on X. The main goal is to determine, for each del Pezzo surface of degree d ≤ 5, which curve classes D (of anticanonical degree ≤ 2d) give rise to an ample, nef‑big, or merely nef (but not big) orbifold divisor for various values of ε.

The paper begins with a concise review of Campana’s orbifold framework and explains why the positivity of −(K_X + ε D) is the key condition for a del Pezzo orbifold. The authors then recall the classical description of del Pezzo surfaces: for degree d they are either P² blown up at 9 − d general points, P¹ × P¹, or a double cover of P² branched over a smooth quartic (degree 2). They collect elementary tools—adjunction, the Riemann–Roch theorem, the Hodge index theorem, and Kawamata–Viehweg vanishing—to control the geometry of curves on these surfaces.

A central technical step is the enumeration of all irreducible curves D on a del Pezzo surface X with anticanonical degree m = −K_X·D ≤ 2d. Using the representation D ∼ a H − ∑ b_i E_i (where H is the pull‑back of a line in P² and the E_i are exceptional divisors of a suitable blow‑up), the authors translate the conditions m and n = D² into a quadratic Diophantine equation in a. By applying Cauchy–Schwarz and the Hodge index inequality they bound the coefficients b_i, then solve the resulting quadratic to obtain a finite list of possible (a, b_i). This computation is implemented in a Python script based on Harbourne’s method, and the resulting curve classes are cross‑checked by intersecting them with all (−1)-curves on X.

With the list of candidate curves in hand, the paper analyses the positivity of −(K_X + ε D). For each degree d the authors distinguish several families of D:

Degree 1: D may be a (−1)-curve (240 possibilities) or a member of the anticanonical linear system |−K_X| (self‑intersection 1). If D∈|−K_X| then −(K_X + ε D) is ample for every 0 < ε < 1. If D∈|−2K_X| and ε = ½, the divisor is nef but not big; otherwise it fails to be nef.

Degree 2: D∈|−K_X| gives ampleness for all ε∈(0,1). A (−1)-curve yields a nef‑big divisor when ε = ½. If D is a smooth fiber of a conic fibration or belongs to |−2K_X|, the divisor is only nef (ε = ½) and not big.

Degree 3: The same pattern holds with additional possibilities: pull‑backs of the hyperplane class from P² (via the blow‑up) and smooth fibers of conic fibrations give nef‑big divisors at ε = ½, while pull‑backs of anticanonical divisors from a degree‑4 del Pezzo surface or members of |−2K_X| give merely nef divisors.

Degree 4: Besides (−1)-curves, smooth conic fibers and anticanonical members are ample for any ε∈(0,1). At ε = ½, pull‑backs of the hyperplane class from P² or the (1,1) class from P¹ × P¹ are nef‑big; divisors linearly equivalent to −K_X + F (F a conic) or members of |−2K_X| are nef but not big.

Degree 5: The richest case. D may be a (−1)-curve, a conic fiber, a pull‑back of a line in P², or any anticanonical member, all giving ampleness for ε∈(0,1). At ε = ½, pull‑backs of the (1,1) class from P¹ × P¹, the class 2H′ − E (where H′ is the hyperplane class after blowing up one point), anticanonical members of a smooth sextic del Pezzo surface, or divisors of the form −K_X + F are nef‑big. Finally, pull‑backs of (1,2) or (2,1) classes from P¹ × P¹, as well as certain linear combinations such as 2H or 4H − 2E_i − 2E_j − 2E_k, give nef but non‑big divisors when ε = ½.

The authors summarize these results in Theorem 1.2, providing a clear checklist for each degree d. They also include a table of self‑intersection numbers for all irreducible curves of anticanonical degree ≤ 2d, which serves as a reference for future work.

Methodologically, the paper demonstrates that a combination of classical surface theory tools suffices to achieve a complete classification without invoking deep results from the Minimal Model Program. The explicit nature of the classification makes it immediately applicable to the study of Campana rational connectedness: the existence of a free Campana curve (a curve whose class lies in the interior of the nef cone) on a low‑degree del Pezzo surface can now be checked case‑by‑case using the tables provided.

In the concluding remarks, the author notes that the classification of weak del Pezzo orbifolds (where −(K_X + ε D) is merely nef) is also obtained, and suggests extensions to higher degree surfaces, multi‑component boundaries, and positive characteristic settings. The work thus fills a gap in the literature on orbifold geometry of rational surfaces and lays a concrete foundation for further arithmetic applications involving Campana points and weak approximation.