For an nn-cyclic covering π ⁣:X→P2\pi\colon X\to\mathbb{P}^2 branched over a generic smooth plane curve of degree nbnb with n>9n>9, Fuentes García (2008) asked whether the Seshadri constant of π∗OP2(1)\pi^*\mathcal{O}_{\mathbb{P}^2}(1) at a very general point of XX attains the maximal value n\sqrt n allowed by the general bound ε≤L2\varepsilon\le\sqrt{L^2}. The source settled the perfect-square case (where ε=n\varepsilon=\sqrt n follows from its bounds) and computed the constants at ramification points for n≤9n\le9, translating the general question into a Nagata-type problem about plane curves passing with prescribed multiplicities through clusters of infinitely near points attached to the branch curve. The question remains open. The closest result (Küronya–Rollenske, 2022) bounds the constant at a general point to the interval [n−n/b,n][\sqrt n-n/b,\sqrt n] for b≥3b\ge3. Unconditional examples of irrational Seshadri constants on surfaces are now known (Laface–Ugaglia, arXiv:2609.26521, realizing ε=25\varepsilon=2\sqrt5 on the blow-up of P2\mathbb{P}^2 at nine very general points, extended to an infinite series by Malara–Merta–Szpond–Zieliński, arXiv:2610.01783); an affirmative answer for any non-square n>9n>9 would settle Fuentes García's conjecture and produce irrational Seshadri constants in this explicit family of cyclic coverings.

Background

Seshadri constants, introduced by J.-P. Demailly, measure the local positivity of a line bundle. For a nef line bundle LL (one whose intersection with every curve is non-negative, L⋅C≥0L\cdot C\ge 0) on a smooth complex projective surface XX and a point x∈Xx\in X, the Seshadri constant of LL at xx is ε(L,x)=inf⁡CL⋅Cmult⁡xC,\varepsilon(L,x)=\inf_{C}\frac{L\cdot C}{\operatorname{mult}_x C}, where CC runs over reduced irreducible curves through xx and mult⁡xC\operatorname{mult}_x C is the multiplicity of CC at xx; for several points P1,…,PrP_1,\dots,P_r the denominator is replaced by the sum of the multiplicities (multiple Seshadri constants). A standard consequence of the Hodge index theorem gives the universal upper bound ε(L,x)≤L2\varepsilon(L,x)\le\sqrt{L^2}, and when the bound is not attained, a reduced irreducible curve computing the constant exists; such a curve is called a Seshadri exceptional curve. Every Seshadri constant computed before 2026 was rational, and whether any polarized surface admits an irrational Seshadri constant was long one of the basic open problems of the subject; examples are now known (Laface–Ugaglia, arXiv:2609.26521; Malara–Merta–Szpond–Zieliński, arXiv:2610.01783).

Fuentes García (2008) approached this problem through cyclic coverings. An nn-cyclic covering is a finite morphism π ⁣:X→Y\pi\colon X\to Y of smooth surfaces built from a line bundle MM on YY and a smooth branch divisor B∼nMB\sim nM; the covering surface XX carries an automorphism σ\sigma of order nn (cyclically permuting the nn sheets) whose fixed locus is the ramification divisor R=π−1(B)R=\pi^{-1}(B). Curves on XX that are invariant under σ\sigma and have high multiplicity at points of RR are natural candidates for Seshadri exceptional curves. Using them, Fuentes García extended a theorem of Steffens (1998): if the base YY has Picard number one (its Néron–Severi space, the group of divisors modulo numerical equivalence tensored with Q\mathbb{Q}, has dimension one) and LL is the ample generator, then at very general points — points avoiding countably many proper algebraic subsets — the multiple Seshadri constants of π∗L\pi^*L are squeezed between a rounded rational value and the square-root bound, with equality on the right whenever a certain square root is an integer. For Y=P2Y=\mathbb{P}^2, L=OP2(1)L=\mathcal{O}_{\mathbb{P}^2}(1) and a branch curve of degree nbnb one has (π∗L)2=n(\pi^*L)^2=n, so at a very general point η∈X\eta\in X ⌊n⌋≤ε(π∗L,η)≤n,\lfloor\sqrt n\rfloor\le\varepsilon(\pi^*L,\eta)\le\sqrt n, with ε=n\varepsilon=\sqrt n whenever nn is a perfect square. For 2≤n≤92\le n\le9 he determined the constants at very general points of the ramification divisor (the values 1,3/2,2,2,12/5,21/8,48/17,31,3/2,2,2,12/5,21/8,48/17,3 for n=2,…,9n=2,\dots,9; the case n=8n=8 requires the branch divisor to be generic), while at a very general point of XX only the interval between that value and n\sqrt n is controlled.

The structural insight connecting the covering problem to plane-curve singularities is that an invariant exceptional divisor D=π∗CD=\pi^*C through a ramification point corresponds to a plane curve CC passing through a cluster of nn infinitely near points — a chain Q1,…,QnQ_1,\dots,Q_n of points on successively blown-up surfaces, each lying on the strict transform of the branch curve — with multiplicities summing to mult⁡(D)\operatorname{mult}(D). Thus exceptional curves on the cover encode plane curves of given degree with imposed singularities at a special cluster. This is a close relative of the Nagata conjecture (1959), which asserts that for n≥10n\ge10 general points P1,…,Pn∈P2P_1,\dots,P_n\in\mathbb{P}^2 any curve of degree dd with multiplicities kik_i at the PiP_i satisfies d≥1n∑ikid\ge\frac{1}{\sqrt n}\sum_i k_i, equivalently ε(OP2(1);n)=1/n\varepsilon(\mathcal{O}_{\mathbb{P}^2}(1);n)=1/\sqrt n; it is known for n≤9n\le9 and for nn a perfect square, and open otherwise. In the covering problem the cluster is determined by the branch curve, so the role of "general position of the points" is played by genericity of the branch divisor (the branch curve ranging over a Zariski-open subset of its linear system), and the two problems are related but not equivalent. Maximality at a very general point is not automatic even in low degrees: for the double plane branched over a conic (n=2n=2, b=1b=1) the cover is P1×P1\mathbb{P}^1\times\mathbb{P}^1 and ε=1<2\varepsilon=1<\sqrt2, a degeneration the source itself records.

Problem Statement

Let n>9n>9 and b≥1b\ge1 be integers, let π ⁣:X→P2\pi\colon X\to\mathbb{P}^2 be the nn-cyclic covering whose branch divisor is a generic smooth plane curve of degree nbnb, and let η∈X\eta\in X be a very general point. Decide whether the Seshadri constant of L=π∗OP2(1)L=\pi^*\mathcal{O}_{\mathbb{P}^2}(1) (so L2=nL^2=n) at η\eta attains the maximal value ε(π∗OP2(1),η)=n,\varepsilon\bigl(\pi^*\mathcal{O}_{\mathbb{P}^2}(1),\eta\bigr)=\sqrt n, as conjectured by Fuentes García. Equivalently: show that there is no Seshadri exceptional (submaximal) curve at a very general point, that is, no reduced irreducible curve C⊂XC\subset X with C⋅L/mult⁡ηC<nC\cdot L/\operatorname{mult}_\eta C<\sqrt n. Since the source's bounds already give ε=n\varepsilon=\sqrt n whenever nn is a perfect square, the open content is the case of non-square n>9n>9, where an affirmative answer would settle Fuentes García's conjecture for this explicit family of cyclic coverings. By the source's Proposition 5.3, invariant exceptional divisors on the cover through a ramification point correspond to plane curves of given degree passing with prescribed multiplicities through the cluster of nn infinitely near points attached to the branch curve, so the conjecture is closely related to a Nagata-type lower bound for linear systems with imposed infinitely near singularities along a generic branch curve.

References

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