General versus very general points for multi-point Seshadri constants on surfaces

For a smooth complex projective surface $X$ and a divisor $L$ that is big and nef (numerically effective, with positive self-intersection), the weighted multi-point Seshadri constant $\varepsilon(X,L,l_1p_1,\dots,l_np_n)$ measures the local positivity of $L$ at $n$ points. Its supremum $\varepsilon(X,L,n,\ell)$ over all configurations of $n$ distinct points equals the value at very general configurations and is bounded above by $\sqrt{L^2/\ell^2}$. Harbourne and Roé proved that whenever the constant is strictly submaximal, $\varepsilon(X,L,n,\ell)<\sqrt{L^2/\ell^2}$, the supremal value is attained on a nonempty Zariski-open set of configurations. The open problem is the remaining maximal case: when $\varepsilon(X,L,n,\ell)=\sqrt{L^2/\ell^2}$, is the value still attained at general rather than merely very general points? Even for the projective plane with homogeneous weights and $n$ a perfect square at least 16, where the value $1/\sqrt n$ is known by Nagata's theorem, this is open; for non-square $n\ge10$ it is the attainment-strengthening companion of Nagata's conjecture.

Background

Let $X$ be a smooth complex projective surface and $L$ a divisor on $X$. The Seshadri constant, a measure of the local positivity of $L$, is defined at distinct points $p_1,\dots,p_n\in X$ with positive integer weights $l_1,\dots,l_n$ by

$$\varepsilon(X,L,l_1p_1,\dots,l_np_n)=\inf_C\frac{L\cdot C}{\sum_{i=1}^n l_i\,\mathrm{mult}_{p_i}C},$$

where the infimum is taken over all reduced irreducible curves $C\subset X$ passing through at least one of the points, $\mathrm{mult}_{p_i}C$ is the multiplicity of $C$ at $p_i$, and $L\cdot C$ is the intersection number. The classical case is $L$ an ample line bundle with all weights equal to 1. Blowing up the points, $\pi:Y\to X$, where $E_i$ denotes the class of the exceptional curve over $p_i$, gives an equivalent characterization: $\varepsilon(X,L,l_1p_1,\dots,l_np_n)$ is the largest real $t$ such that the divisor $\pi^*L-t(l_1E_1+\dots+l_nE_n)$ is nef (numerically effective, that is, having non-negative intersection with every curve). Since a nef divisor on a surface has non-negative self-intersection, every configuration satisfies

$$\varepsilon(X,L,l_1p_1,\dots,l_np_n)\le\sqrt{L^2/\ell^2},\qquad \ell^2=l_1^2+\dots+l_n^2,$$

which is therefore called the expected or maximal value. A curve with $L\cdot C/(\sum_i l_i\,\mathrm{mult}_{p_i}C)<\sqrt{L^2/\ell^2}$ is called abnormal (Nagata's terminology) or submaximal.

Write $\varepsilon(X,L,n,\ell)$ for the supremum of $\varepsilon(X,L,l_1p_1,\dots,l_np_n)$ over all configurations of $n$ distinct points. A standard semicontinuity argument shows that this supremum equals the value at any very general configuration, meaning one that avoids a countable union of proper Zariski-closed subsets of the configuration space. The strictly weaker property of being general, meaning lying in a single nonempty Zariski-open subset, is what makes a statement hold on an honest open set of configurations rather than only on a countable intersection of open sets; whether the supremal value is attained on a Zariski-open set is the question at issue.

The most famous instance is Nagata's conjecture (1959): for $X=\mathbf{P}^2$, $L$ a line, and homogeneous weights, the constant $\varepsilon(n):=\varepsilon(\mathbf{P}^2,\mathcal{O}(1),n)$ should equal its maximal value $1/\sqrt n$ for $n\ge10$ very general points; equivalently, the linear system of plane curves of degree $d$ having multiplicity at least $m$ at each of the $n$ points is empty whenever $d<m\sqrt n$. Nagata proved the conjecture when $n$ is a perfect square, and it remains open for every non-square $n\ge10$. In this language, the conjecture asserts that no abnormal curve exists at a very general configuration of $n\ge10$ points of $\mathbf{P}^2$.

Harbourne and Roé (2008) proved a discreteness theorem: for fixed $(X,L,n,\ell)$ with $L$ big (on a surface, $L^2>0$) and nef, the values of $\varepsilon$ at general configurations form a discrete subset of the real line, apart from the single possible limit point $\sqrt{L^2/\ell^2}$. A direct consequence is that whenever the constant is strictly submaximal, $\varepsilon(X,L,n,\ell)<\sqrt{L^2/\ell^2}$, there is a nonempty Zariski-open set of configurations attaining the supremal value: in that regime an abnormal curve at a general configuration must have its class in a finite explicit obstruction set, and excluding finitely many classes removes only Zariski-closed subsets. Earlier results — Oguiso's study of Seshadri constants in families of surfaces, and the survey of Strycharz-Szemberg and Szemberg, which already asked for square numbers of points whether the hypothesis that the points are very general can be relaxed to general — suggested that open-set attainment should hold in general. This is exactly what the discreteness mechanism cannot deliver in the maximal case, where the finite obstruction sets degenerate at the limit point $\sqrt{L^2/\ell^2}$.

Problem Statement

Let $X$ be a smooth complex projective surface, $L$ a big and nef $\mathbf{Q}$-divisor on $X$ (in particular, an ample line bundle), $n\ge1$ an integer, and $\ell=(l_1,\dots,l_n)$ a tuple of positive integers. Let $\varepsilon(X,L,n,\ell)$ denote the supremum of the weighted multi-point Seshadri constant over all configurations of $n$ distinct points of $X$; recall that $\varepsilon(X,L,n,\ell)$ equals the value at very general configurations and that $\varepsilon(X,L,l_1p_1,\dots,l_np_n)\le\sqrt{L^2/\ell^2}$ for every configuration, with equality possible.

Harbourne and Roé proved that if $\varepsilon(X,L,n,\ell)<\sqrt{L^2/\ell^2}$, then there exists a nonempty Zariski-open subset $U$ of the configuration space of $n$ distinct points such that $\varepsilon(X,L,l_1p_1,\dots,l_np_n)=\varepsilon(X,L,n,\ell)$ for every configuration in $U$.

Problem. Does the same conclusion hold in the maximal case? That is, when $\varepsilon(X,L,n,\ell)=\sqrt{L^2/\ell^2}$, does there exist a nonempty Zariski-open subset $U$ of the configuration space of $n$ distinct points such that

$$\varepsilon(X,L,l_1p_1,\dots,l_np_n)=\varepsilon(X,L,n,\ell)=\sqrt{L^2/\ell^2}$$

for every configuration $(p_1,\dots,p_n)\in U$? Equivalently: is the very-general value of a multi-point Seshadri constant on a surface always attained at general points?

No restriction to homogeneous weights, to surfaces of Picard rank one (surfaces on which all curve classes are numerically proportional), or to a particular class of surfaces is intended. Two concrete instances calibrate the problem: for $(\mathbf{P}^2,\mathcal{O}(1))$ with homogeneous weights and $n$ a perfect square at least 16, the very-general value $1/\sqrt n$ is known by Nagata's theorem but open-set attainment is open; for non-square $n\ge10$, Nagata's conjecture predicts maximality and the present question is the complementary attainment statement — the two assertions do not imply one another.

References

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