How are low degree del Pezzo orbifolds classified by their boundary components?

A del Pezzo orbifold adds weights to a boundary

A del Pezzo orbifold is a Campana orbifold (X, Δ_ε) in which X is a smooth del Pezzo surface and the Q-divisor −(K_X + Δ_ε) is ample. The boundary Δ is a reduced strict normal crossings divisor with irreducible components D_1, …, D_r, and each component carries a multiplicity m_i ≥ 2 and weight ε_i = 1 − 1/m_i, so Δ_ε = Σ ε_i D_i. The degree of the orbifold is the degree d = K_X² of the underlying surface, not a count of boundary components.

The boundary is called irreducible when r = 1 and reducible when r ≥ 2. The earlier work of the author handled the irreducible case; this paper treats the complementary reducible case, and the two together complete the classification for degree at most five. Because every weight satisfies ε_i ≥ 1/2, the paper records a necessary condition: −(K_X + ½Δ) must be nef, where Δ is the reduced boundary.

Reducible boundaries create new multiplicity constraints

With an irreducible boundary, the multiplicity of the single component is the only weight to choose. With several components, the weights interact through the (−1)-curves that meet more than one component, and this produces arithmetic constraints on the multiplicities that have no analogue in the irreducible case. The paper's Example 1.3 makes the interaction concrete on a smooth cubic surface: for three coplanar lines ℓ_1, ℓ_2, ℓ_3 with ℓ_1 + ℓ_2 + ℓ_3 in |−K_X|, the pair is a del Pezzo orbifold exactly when 1/m_1, 1/m_2, 1/m_3 satisfy the strict triangle inequalities.

That example also shows the admissible multiplicity vectors are not closed under lowering multiplicities: (3, 4, 11) is admissible while (2, 4, 11) is not. A second difference is that the strict normal crossings condition is no longer automatic, so the answer can depend on where X sits in its moduli space. For the three coplanar lines, the divisor is SNC precisely when the lines are not concurrent, that is, when they do not meet in an Eckardt point.

The reduced boundary has anticanonical degree at most 2d−1

A short observation bounds the search. Since −K_X and −(K_X + Δ_ε) are both ample, their intersection number is positive, giving Σ ε_i (−K_X · D_i) < d. With ε_i ≥ 1/2 and −K_X · D_i ≥ 1 for every component, the reduced boundary satisfies −K_X · Δ ≤ 2d − 1, and in particular r ≤ 2d − 1. So for each fixed degree the underlying pairs (X, Δ) form a bounded family, even though the multiplicities themselves need not be bounded.

This bound is what makes a case-by-case classification feasible. For each d in {1, 2, 3, 4, 5} the paper determines three things: the curves that can occur as components, described through the classification of curves of low anticanonical degree; the possible configurations of the boundary, recorded by the classes of the components and their mutual intersection numbers; and, for each configuration, the set of all multiplicity choices m_i ≥ 2 for which −(K_X + Δ_ε) is ample.

In degree one only two anticanonical members survive

For a del Pezzo surface S of degree 1, the paper first records how each possible type of component meets the (−1)-curves. A (−1)-curve component has self-intersection −1, is disjoint from 56 of the (−1)-curves, meets 126 of them at one point, meets 56 of them twice, and meets exactly one (−1)-curve with intersection number 3. A member of the anticanonical class |−K_S| meets every (−1)-curve once. A smooth fibre of a conic fibration has 14 (−1)-curves with intersection number 0, 64 with 1, 84 with 2, 64 with 3, and 14 with 4.

Theorem 2.1 then states that −(K_S + Δ_ε) is nef if and only if Δ_ε = ½D_1 + ½D_2, where D_1 and D_2 are both members of the anticanonical class |−K_S|. The proof rules out the alternatives by exhibiting (−1)-curves with intersection number at least 2 against the second component: a (−1)-curve, a smooth conic fibre, or a member of |−2K_S| each admit such a curve, and the pullback of an anticanonical member from a degree 2 del Pezzo surface admits more than one. A conic-fibre component is excluded outright because 78 (−1)-curves meet it with intersection number at least 3.

Degree four has five orbifold types but 106 nef reduced curves

For a smooth del Pezzo surface S of degree 4, the paper fixes a birational morphism to P² contracting disjoint (−1)-curves E_1, …, E_5, with H the pullback of a line class and −K_S = 3H − E_1 − ⋯ − E_5. The sixteen (−1)-curves, called lines, are the E_i, the L_ij = H − E_i − E_j, and L_0 = 2H − E_12345. Two lines with intersection number 1 are neighbours, and the Weyl group W(D_5) of order 1920 acts on the lines as the full automorphism group of their intersection graph, the Clebsch graph.

Theorem 5.3 classifies all del Pezzo orbifolds with r ≥ 2: there are only five types, and for each the admissible weights are determined. Separately, Theorem 5.6 lists all reduced curves D with r ≥ 2 components for which −(K_S + ½D) is nef, and there are 106 such types. The two counts answer different questions: five is the number of orbifold types with ample log anticanonical divisor, while 106 counts reduced boundaries that only pass the weaker nefness test at the uniform weight ½. The paper notes that the 106 types arise from a case-by-case analysis organised by the component of largest degree.

Degree five yields ten boundary orbits with explicit weight inequalities

For a quintic del Pezzo surface, the automorphism group is S_5 and the paper works up to that action. Table 8 lists 23 S_5-orbits of classes of irreducible curves of anticanonical degree at most 9, labelled T1 through T23, with each representative's anticanonical degree, self-intersection, arithmetic genus, and intersection profile against the ten lines. Table 9 then lists the boundaries D = D_1 + ⋯ + D_r with r ≥ 2 that admit weights making −(K_S + Δ_ε) ample, giving ten orbits.

The last column of Table 9 records only the ampleness conditions that are not automatic for weights in [½, 1). Several entries have no conditions at all, marked with a dagger, meaning m_i = 2 for every component is admissible; these include the two-component cases E_1 with H − E_1 − E_4, and E_1 with H − E_1. Other entries carry genuine inequalities, such as −ε_1 + 2ε_2 < 1 for the pair E_1 and 3H − 2E_1 − E_2 − E_3 − E_4, or ε_1 + ε_2 − ε_3 < 1 for the three lines E_1, E_2, H − E_1 − E_2. The column also reports the number of multisets of classes in each orbit, ranging from 5 to 30.

A Python crosscheck reproduces the ten orbits independently

The degree 5 results are verified by Python 3 scripts that prove Theorem 6.6 independently of the earlier section, using only the paper's Lemmas 6.3, 6.4 and Remark 6.1. The first script builds Pic(S), the ten lines and the Weyl group W(A_4), checking order 120 and that it preserves K_S and the lines. It then lists all classes of degree at most 9 that are lines, conic classes, or nef with positive self-intersection, finding 346 classes in 23 orbits.

The second script examines all 92,602 multisets of total degree at most 9 with at least two members, runs Fourier-Motzkin elimination in all weights, and obtains the same ten orbits, with no unresolved case. The author states that the scripts were generated with Claude Opus 5 and 5.5 for crosschecking, organising the tables, and proofreading, while the idea of writing the programs and responsibility for correctness rest with the author. The classification itself is a statement about which boundaries and weights exist; it does not by itself establish Campana rational connectedness or the arithmetic conjectures the introduction connects to these orbifolds.

The decisive split is reducible versus irreducible, not degree alone

The paper's contribution is a completion, and the completion is specifically about reducibility. For degree at most five, the irreducible case was settled earlier; this paper supplies the reducible case, so the two together cover all del Pezzo orbifolds of degree at most five. Readers comparing orbifolds should therefore treat the number of boundary components as the primary axis, because that is where the new constraints come from: multiplicities of different components interact through the (−1)-curves meeting several of them, and the SNC condition can depend on the position of X in its moduli space.

The most decision-relevant distinction inside the paper is between the five degree 4 orbifold types of Theorem 5.3 and the 106 reduced curves of Theorem 5.6, and between the ten degree 5 boundary orbits of Table 9 and the 23 curve orbits of Table 8. These pairs count different objects under different conditions, and the weight inequalities in Table 9 are the part that actually decides whether a given multiplicity vector is admissible. When a configuration appears with no listed condition, m_i = 2 for all components is already enough; when an inequality is listed, lowering a multiplicity can move a configuration out of the admissible set, as the (3, 4, 11) versus (2, 4, 11) example shows.

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